Integrals Class 12th Mathematics Part Ii CBSE Solution

Class 12^th Mathematics Part Ii CBSE Solution

Exercise 7.1

1. sin 2x Find an anti-derivative (or integral) of the following functions by the…
2. cos 3x Find an anti-derivative (or integral) of the following functions by the…
3. e2x Find an anti-derivative (or integral) of the following functions by the…
4. (ax + b)^2 Find an anti-derivative (or integral) of the following functions by…
5. sin 2x - 4 e3x Find an anti-derivative (or integral) of the following functions…
6. Find the following integrals. integrate (4e^3x + 1) dx
7. integrate x^2 (1- 1/x^2) dx Find the following integrals.
8. integrate (ax^2 + bx+c) dx Find the following integrals.
9. integrate 2x^2 + e^x dx Find the following integrals.
10. integrate (root x - 1/root x)^2 dx Find the following integrals.
11. integrate x^3 + 5x^2 - 4/x^2 dx Find the following integrals.
12. integrate x^3 + 3x+4/root x dx Find the following integrals.
13. integrate x^3 - x^2 + x-1/x-1dx Find the following integrals.
14. integrate (1-x) root xdx Find the following integrals.
15. integrate root x (3x^2 + 2x+3) dx Find the following integrals.
16. integrate (2x-3cosx+e^x) dx Find the following integrals.
17. integrate (2x^2 - 3sinx+5 root x) dx Find the following integrals.…
18. integrate secx (secx+tanx) dx Find the following integrals.
19. integrate sec^2x/cosec^2xdx Find the following integrals.
20. integrate 2-3sinx/cos^2xdx Find the following integrals.
21. The anti-derivative of (root x + 1/root x) equalsA. 1/3 x^1/3 + 2x^1/2 + c B.…
22. If d/dx f (x) = 4x^3 - 3/x^4 such that f(2) = 0. Then f(x) isA. x^4 + 1/x^3 -…

Exercise 7.10

1. Evaluate the integrals using substitution. integrate _0^1 x/x^2 + 1 dx…
2. integrate _0^pi /2 root sinphi cos^5phi d phi Evaluate the integrals using…
3. integrate _0^1sin^-1 (2x/1+x^2) dx Evaluate the integrals using substitution.…
4. integrate _0^2root x+2 (putx+2 = t^2) Evaluate the integrals using substitution.…
5. integrate _0^pi /2 sinx/1+cos^2xdx Evaluate the integrals using substitution.…
6. integrate _0^2 dx/x+4-x^2 Evaluate the integrals using substitution.…
7. integrate _-1^1 dx/x^2 + 2x+5 Evaluate the integrals using substitution.…
8. integrate _1^2 (1/x - 1/2x^2) e^2x dx Evaluate the integrals using substitution.…
9. The value of the integral integrate _ 1/3^1 (x-x^3)^1/3/x^4 dx isA. 6 B. 0 C. 3…
10. If f (x) = integrate _0^xtsintegrate dt then f'(x) isA. B. xsinx C. xcosx D.…

Exercise 7.11

1. integrate _0^pi /2 cos^2xdx By using the properties of definite integrals,…
2. integrate _0^pi /2 root sinx/root sinx + root cosx dx By using the properties of…
3. integrate _0^pi /2 sin^3/2 xdx/sin^3/2 x+cos^3/2 x dx By using the properties of…
4. integrate _0^pi /2 cos^5xdx/sin^5x+cos^5x By using the properties of definite…
5. integrate _-5^5 |x+2|dx By using the properties of definite integrals, evaluate…
6. integrate _2^8 |x-5|dx By using the properties of definite integrals, evaluate…
7. integrate _0^1z (1-x)^n dx By using the properties of definite integrals,…
8. integrate _0^pi /4 log (1+tanx) dx By using the properties of definite…
9. integrate _0^2x root 2-xdx By using the properties of definite integrals,…
10. integrate _0^pi /2 (2logsinx-logsin2x) dx By using the properties of definite…
11. integrate _ - pi /2^pi /2 sin^2xdx By using the properties of definite…
12. integrate _0^pi xdx/1+sinx By using the properties of definite integrals,…
13. integrate _ - pi /2^pi /2 sin^7xdx By using the properties of definite…
14. By using the properties of definite integrals, evaluate the integrals integrate…
15. integrate _0^pi /2 sinx-cosx/1+sinxcosxdx By using the properties of definite…
16. integrate _0^pi log (1+cosx) dx By using the properties of definite integrals,…
17. integrate _0^a root x/root x + root a-x dx By using the properties of definite…
18. integrate _0^4 |x-1|dx By using the properties of definite integrals, evaluate…
19. Show that integrate _0^af (x) g (x) dx = 2 integrate _0^af (x) dx_s if f and g…
20. The value of integrate _ - pi /2^pi /2 (x^3 + xcosx+tan^3x+1) dx A. 0 B. 2 C.…
21. The value of integrate _0^pi /2 log (4+3sinx/4+3cosx) dxjs A. 2 B. 3/4 C. 0 D.…

Miscellaneous Exercise

1. Integrate the function: 1/x-x^3
2. Integrate the function: 1/root x+a + root x+b
3. Integrate the function: 1/x root ax-x^2 [: putx = a/t]
4. Integrate the function: 1/x^2 (x^4 + 1)^3/4
5. Integrate the function: 1/x^1/2 + x^1/3 [hintegrate 1/x^1/2 + x^1/3 = 1/x^1/3…
6. Integrate the function: 5x/(x+1) (x^2 + 9)
7. Integrate the function: sinx/sin (x-a)
8. Integrate the function: e^5logx-e^4logx/e^3logx-e^2logx
9. Integrate the function: cosx/root 4-sin^2x
10. Integrate the function:
11. Integrate the function: 1/cos (x+a) cos (x+b)
12. Integrate the function: x^3/root 1-x^8
13. Integrate the function: e^x/(1+e^x) (2+e^x)
14. Integrate the function: 1/(x^2 + 1) (x^2 + 4)
15. Integrate the function: cos^3xe^logsinx
16. Integrate the function: e^3logx (x^4 + 1)^-1
17. Integrate the function: f^there there eξ sts (ax+b) [f (ax+b)]^n
18. Integrate the function: 1/root sin^3xsin (x + alpha)
19. Integrate the function: sin^-1root x-cos^-1root x/sin^-1root x+cos^-1root x , x…
20. Integrate the function: root 1 - root x/1 + root x
21. Integrate the function: 2+sin2x/1+cos2xe^x
22. Integrate the function: x^2 + x+1/(x+1)^2 (x+2)
23. Integrate the function: tan^-1root 1-x/1+x
24. Integrate:
25. Evaluate the definite integral: integrate _ pi /2^pi e^x (1-sinx/1-cosx) dx…
26. Evaluate the definite integral: integrate _0^pi /4 sinxcosx/cos^4x+sin^4xdx…
27. Evaluate the definite integral: integrate _0^pi /2 cos^2xdx/cos^2x+4sin^2x…
28. Evaluate the definite integral: integrate _ pi /6^pi /3 sinx+cosx/root sin2x dx…
29. Evaluate the definite integral: integrate _0^1 dx/root 1+x - root x…
30. Evaluate the definite integral: integrate _0^pi /4 sinx+cosx/9+16sin2xdx…
31. Evaluate the definite integral: integrate _0^pi /2 sin2xtan^-1 (sinx) dx…
32. Evaluate the definite integral: integrate _0^pi xtanx/secx+tanxdx…
33. Evaluate the definite integral: integrate _1^4 [|x-1|+|x-2|+|x-3|]dx…
34. Prove: integrate _1^3 dx/x^2 (x+1) = 2/3 + log 2/3
35. Prove: integrate _0^1xe^x dx = 1
36. Prove: integrate _-1^1x^17cos^4xdx = 0
37. Prove: integrate _0^pi /2 sin^3xdx = 2/3
38. Prove: integrate _0^pi /4 2tan^3xdx = 1-log2
39. Prove: integrate _0^1sin^-1xdx = pi /2 - 1
40. Evaluate integrate _0^1e^2-3x dx as a limit of a sum.
41. integrate dx/e^x + e^-x is equal to Choose the correct answersA. log (e^x -…
42. integrate cos2x/(sinx+cosx)^2 dx is equal to Choose the correct answersA.…
43. If f (a + b - x) = f (x), then integrate _a^bxf (x) is equal to Choose the…
44. The value of integrate _0^1tan^-1 (2x-1/1+x-x^2) dx Choose the correct…

Exercise 7.2

1. 2x/1+x^2 Integrate the functions.
2. (logx)^2/x Integrate the functions.
3. 1/x+xlogx Integrate the functions.
4. sin x sin (cos x) Integrate the functions.
5. Integrate the functions. sin (ax + b) cos (ax + b)
6. root ax+b Integrate the functions.
7. x root x+2 Integrate the functions.
8. x root 1+2x^2 Integrate the functions.
9. (4x+2) root x^2 +x+1 Integrate the functions.
10. 1/x - root x Integrate the functions.
11. x/root x+4 , x0 Integrate the functions.
12. (x^3 - 1)^1/x x^5 Integrate the functions.
13. x^2/(2+3x^3)^3 Integrate the functions.
14. 1/x (logx)^m , x0 , m not equal 1 Integrate the functions.
15. x/9-4x^2 Integrate the functions.
16. e^2x+3 Integrate the functions.
17. x/e^x^2 Integrate the functions.
18. e^tan^-1x/1+x^2 Integrate the functions.
19. e^2x - 1/e^2x + 1 Integrate the functions.
20. e^2x - e^-2x/e^2x + e^-2x Integrate the functions.
21. tan^2 (2x - 3) Integrate the functions.
22. sec^2 (7 - 4x) Integrate the functions.
23. sin^-1x/root 1-x^2 Integrate the functions.
24. 2cosx-3sinx/6cosx+4sinx Integrate the functions.
25. 1/cos^2x (1-tanx)^2 Integrate the functions.
26. cosroot x/root x Integrate the functions.
27. root sin2x cos2x Integrate the functions.
28. cosx/root 1+sinx Integrate the functions.
29. cot x log sin x Integrate the functions.
30. sinx/1+cosx Integrate the functions.
31. sinx/(1+cosx)^2 Integrate the functions.
32. 1/1+cotx Integrate the functions.
33. 1/1-tanx Integrate the functions.
34. root tanx/sinxcosx Integrate the functions.
35. (1+logx)^2/x Integrate the functions.
36. (x+1) (x+logx)^2/x Integrate the functions.
37. x^3sin (tan^-1x^4)/1+x^8 Integrate the functions.
38. Choose the correct answer: A. 10x x^10 + C B. 10x + x^10 + C C. (10x x^10)1 + C…
39. integrate dx/sin^2xcos^2x equalsA. tan x + cot x + C B. tan x - cot x + C C.…

Exercise 7.3

1. sin^2 (2x+5) Find the integrals of the functions.
2. cos2xcos4xcos6x Find the integrals of the functions.
3. sin3xcos4x Find the integrals of the functions.
4. sin^3 (2x+1) Find the integrals of the functions.
5. sin^3xcos^3x Find the integrals of the functions.
6. sinxsin2xsin3x Find the integrals of the functions.
7. sin4xsin8x Find the integrals of the functions.
8. 1-cos/1+cos Find the integrals of the functions.
9. cosx/1+cosx Find the integrals of the functions.
10. sin^4x Find the integrals of the functions.
11. cos^42x Find the integrals of the functions.
12. sin^2x/1+cosx Find the integrals of the functions.
13. cos2x-cos2alpha /cosx-cosalpha Find the integrals of the functions.…
14. cosx-sinx/1+sin2x Find the integrals of the functions.
15. tan^32xsec2x Find the integrals of the functions.
16. tan^4x Find the integrals of the functions.
17. sin^3x+cos^3x/sin^2xcos^2x Find the integrals of the functions.
18. cos2x+2sin^2x/cos^2x Find the integrals of the functions.
19. 1/sinxcos^3x Find the integrals of the functions.
20. cos2x/(cosx+sinx)^2 Find the integrals of the functions.
21. sin^-1 (cosx) Find the integrals of the functions.
22. 1/cos (x-a) cos (x-b) Find the integrals of the functions.
23. integrate sin^2x-cos^2x/sin^2xcos^2xdx A. tanx+cotx+c B. tanx+cosecx+c C.…
24. integrate e^x (1+x)/cos^2 (e^xx) A. -cot (ex^4) + c B. tan (e^x) + c C. tan…

Exercise 7.4

1. 3x^2/x^6 + 1 Integrate the functions.
2. 1/root 1+4x^2 Integrate the functions.
3. 1/root (2-x)^2 + 1 Integrate the functions.
4. 1/root 9-25x^2 Integrate the functions.
5. 3x/1+2x^4 Integrate the functions.
6. x^2/1-x^6 Integrate the functions.
7. x-1/root x^2 - 1 Integrate the functions.
8. x^2/root x^6 + a^6 Integrate the functions.
9. sec^2x/root tan^2x+4 Integrate the functions.
10. 1/root x^2 + 2x+2 Integrate the functions.
11. 1/9x^2 + 6x+5 Integrate the functions.
12. 1/root 7-6x-x^2 Integrate the functions.
13. 1/root (x-1) (x-2) Integrate the functions.
14. 1/root 8+3x-x^2 Integrate the functions.
15. 1/root (x-a) (x-b) Integrate the functions.
16. 4x+1/root 2x^2 + x-3 Integrate the functions.
17. x+2/root x^2 - 1 Integrate the functions.
18. 5x-2/1+2x+3x^2 Integrate the functions.
19. 6x+7/root (x-5) (x-4) Integrate the functions.
20. x+2/root 4x-x^2 Integrate the functions.
21. x+2/root x^2 + 2x+3 Integrate the functions.
22. x+3/x^2 - 2x-5 Integrate the functions.
23. 5x+3/root x^2 + 4x+10 Integrate the functions.
24. Choose the correct answer: integrate dx/x^2 + 2x+2 A. xtan^-1 (x+1) + c B.…
25. Choose the correct answer: integrate dx/root 9x-4x^2 A. 1/9 sin^-1 (9x-8/8) + c…

Exercise 7.5

1. x/(x+1) (x+2) Integrate the rational functions.
2. 1/x^2 - 9 Integrate the rational functions.
3. 3x-1/(x-1) (x-2) (x-3) Integrate the rational functions.
4. x/(x-1) (x-2) (x-3) Integrate the rational functions.
5. 2x/x^2 + 3x+2 Integrate the rational functions.
6. 1-x^2/x (1-2x) Integrate the rational functions.
7. x/(x^2 + 1) (x-1) Integrate the rational functions.
8. x/(x-1)^2 (x+2) Integrate the rational functions.
9. 3x+5/x^3 - x^2 - x+1 Integrate the rational functions.
10. 2x-3/(x^2 - 1) (2x+3) Integrate the rational functions.
11. 5x/(x+1) (x^2 - 4) Integrate the rational functions.
12. x^3 + x+1/x^2 - 1 Integrate the rational functions.
13. 2/(1-x) (1+x^2) Integrate the rational functions.
14. 3x-1/(x+2)^2 Integrate the rational functions.
15. 1/x^4 - 1 Integrate the rational functions.
16. 1/x (x^n + 1) [Hint: multiply numerator and denominator by xn-1 and put xn=t]…
17. cosx/(1-sinx) (2-sinx) [: x = t] Integrate the rational functions.…
18. (x^2 + 1) (x^2 + 2)/(x^2 + 3) (x^2 + 4) Integrate the rational functions.…
19. 2x/(x^2 + 1) (x^2 + 3) Integrate the rational functions.
20. 1/x (x^4 - 1) Integrate the rational functions.
21. 1/(6^x - 1) [Hint: Put ex = t] Integrate the rational functions.
22. Choose the correct answer integrate xdx/(x-1) (x-2) A. log| (x-1)^2/x-2|+c B.…
23. Integrate the rational functions.A. log|x| - 1/2 log (x^2 + 1) + c B. log|x| +…

Exercise 7.6

1. xsinx Integrate the functions.
2. xsin3x Integrate the functions.
3. x^2e^x Integrate the functions.
4. xlogx Integrate the functions.
5. xlog2x Integrate the functions.
6. x^2logx Integrate the functions.
7. xsin^-1x Integrate the functions.
8. xtan^-1x Integrate the functions.
9. xcos^-1x Integrate the functions.
10. (sin^-1x)^2 Integrate the functions.
11. xcos^-1x/root 1-x^2 Integrate the functions.
12. xsec^2x Integrate the functions.
13. tan^-1x Integrate the functions.
14. x (logx)^2 Integrate the functions.
15. (x^2 + 1) logx Integrate the functions.
16. e^x (sinx+cosx) Integrate the functions.
17. xe^x/(1+x)^2 Integrate the functions.
18. e^x (1+sinx/1+cosx) Integrate the functions.
19. e^x (1/x - 1/x^2) Integrate the functions.
20. (x-3) e^x/(x-1)^3 Integrate the functions.
21. e^2xsinx Integrate the functions.
22. sin^-1 (2x/1+x^2) Integrate the functions.
23. integrate x^2e^x^3 dxequdls Choose the correct answer:A. 1/3 e^x^3 + c B. 1/3…
24. integrate e^xsecx (1+tanx) dx Choose the correct answer:A. e^xcosx+c B.…

Exercise 7.7

1. Integrate : root 4-x^2
2. Integrate : root 1-4x^2
3. Integrate : root x^2 + 4x+6
4. Integrate : root x^2 + 4x+1
5. Integrate : root 1-4x-x^2
6. Integrate : root x^2 + 4x-5
7. Integrate : root 1+3x-x^2
8. Integrate : root x^2 + 3x
9. Integrate : root 1 + x^2/9
10. integrate root 1+x^2 dx is equal to Choose the correct answerA. x/2 root 1+x^2…
11. integrate root x^2 - 8x+7 dx is equal to A. 1/2 (x-4) root x^2 - 2x+7 +…

Exercise 7.8

1. integrate _a^bxdx Evaluate the following definite integrals as limit of sums.…
2. integrate _0^5 (x+1) dx Evaluate the following definite integrals as limit of…
3. integrate _2^3x^2 dx Evaluate the following definite integrals as limit of sums.…
4. integrate _1^4 (x^2 - x) dx Evaluate the following definite integrals as limit…
5. integrate _-1^1e^x dx Evaluate the following definite integrals as limit of…
6. integrate _0^4 (x+e^2x) dx Evaluate the following definite integrals as limit of…

Exercise 7.9

1. Evaluate integrate _-1^1 (x+1) dx
2. Evaluate integrate _2^3 1/x dx
3. Evaluate integrate _1^2 (4x^3 -5x^2 +6x+9) dx
4. Evaluate integrate _0^pi /4 sin2xdx
5. Evaluate integrate _0^pi /2 cos2xdx
6. Evaluate integrate _4^5e^x dx
7. Evaluate integrate _0^pi /4 tanxdx
8. Evaluate integrate _ pi /6^pi /4 cosecxdx
9. Evaluate integrate _0^1 dx/root 1-x^2
10. Evaluate integrate _0^1 dx/1+x^2
11. Evaluate integrate _2^3 dx/x^2 - 1
12. Evaluate integrate _0^pi /2 cos^2xdx
13. Evaluate integrate _2^3 xdx/x^2 + 1
14. Evaluate integrate _0^1 2x+3/5x^2 + 1 dx
15. Evaluate integrate _0^1xe^x^2 dx
16. Evaluate integrate _1^2 5x^2/x^2 + 4x+3
17. Evaluate integrate _0^pi /1 (2sec^2x+x^3 + 2) dx
18. Evaluate integrate _0^pi (sin^2 x/2 - cos^2 x/2) dx
19. Evaluate integrate _0^2 6x+3/x^2 + 4 dx
20. Evaluate integrate _0^1 (xe^x + sin pi x/4) dx
21. integrate _1^root 3 dx/x^2 + 1 equalsA. π/3 B. 2π/3 C. π/6 D. π/12…
22. integrate dx/4+9x^2 equalsA. π/6 B. π/12 C. π/24 D. π/4

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Exercise 7.1

Question 1.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

sin 2x

Answer:

Method: To find the anti derivative of a function by inspection 
Steps: 1. In this method we look for a function whose derivative is the given function. For Example: if we need to find anti derivative of x², we know that derivative of x³ is 3x². Therefore, the variable terms comes out to be the same.
2. After that balance out the coefficients of variables by dividing and multiplying suitable terms. From above example if [we divide x³ by 3 we will get the answer as x². Hence, we can say that anti derivative of x² is x³/3.

Now, similarly,

We know that 

Therefore, the anti-derivative of sin2x is 

Question 2.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

cos 3x

Answer:

We know that 

Therefore, the anti-derivative of cos3x is .

Question 3.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

e^2x

Answer:

We know that 

Therefore, the anti-derivative of  is .

Question 4.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

(ax + b)²

Answer:

We know that 

Therefore, the anti-derivative of  is .

Question 5.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

sin 2x – 4 e^3x

Answer:

We know that

Therefore, the anti-derivative of sin2x is  …(1)

Also,

Therefore, the anti-derivative of  is  …(2)

From (1) and (2), we get,

= sin 2x – 4e^3x

Therefore, the anti-derivative of sin 2x – 4 e^3x is .

Question 6.

Find the following integrals.

Answer:

Question 7.

Find the following integrals.

Answer:

Question 8.

Find the following integrals.

Answer:

Question 9.

Find the following integrals.

Answer:

Question 10.

Find the following integrals.

Answer:

Now we know that,

∫xⁿ dx 

Therefore,

Question 11.

Find the following integrals.

Answer:

Separating the terms we get,

Applying the formula,
∫xⁿ dx = 

Answer. 
Question 12.

Find the following integrals.

Answer:

Separating the terms we get,

Applying the formula,

∫ xⁿ dx = 

Question 13.

Find the following integrals.

Answer:

Now the numerator can be factorized as,

x³ - x² + x - 1 = x²(x - 1) + 1(x - 1)

x³ - x² + x - 1 = (x² + 1)(x - 1)

Now putting this in given integral we get,

Question 14.

Find the following integrals.

Answer:

Question 15.

Find the following integrals.

Answer:

Question 16.

Find the following integrals.

Answer:

Question 17.

Find the following integrals.

Answer:

Question 18.

Find the following integrals.

Answer:

Formulas Used: ∫ sec²x dx = tanx + c and ∫ secx tanx dx = sec x + c

Opening the brackets we get,

Answer. 
Question 19.

Find the following integrals.

Answer:

= tanx –x +C

Question 20.

Find the following integrals.

Answer:

= 2tanx – 3secx + C

Question 21.

The anti-derivative of  equals
A. 

B. 

C. 

D. 

Answer:

Question 22.

If  such that f(2) = 0. Then f(x) is
A.  B. 

C.  D. 

Solution ||| The correct option is (A).

Answer:

It is given that 

Also, It is given that f(2) = 0

Therefore, 

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Exercise 7.10

Question 1.

Evaluate the integrals using substitution.

Answer:

Given: 

Let x² + 1 = t

⇒ 2xdx = dt

⇒ xdx = � dt

When x = 0, t = 1 and when x = 1, t = 2

Question 2.

Evaluate the integrals using substitution.

Answer:

Given: 

Let 

Also, let 

when,

so,

Question 3.

Evaluate the integrals using substitution.

Answer:

Given: 

Let x = tan θ ⇒ dx = sec² θ d θ

When, x = 0, θ = 0 and when x = 1, θ = π /4

Let

By applying product rule as,

Question 4.

Evaluate the integrals using substitution.

Answer:

Given: 

Let x + 2 = t² ⇒ dx = 2t dt

And x = t² -2

when, x = 0, t = √2 and when x = 2, t = 2

so,

Question 5.

Evaluate the integrals using substitution.

Answer:

Given: 

Let cos x = t

⇒ -sin xdx = dt

⇒ sin xdx = -dt

When x = 0, t = 1 and when x = π /2, t = 0

because, 

Question 6.

Evaluate the integrals using substitution.

Answer:

Given: 

we can write it as,

let

when

because,

Question 7.

Evaluate the integrals using substitution.

Answer:

Given: 

Let x + 1 = t

⇒ dx = dt

When x = -1, t = 0 and when x = 1, t = 2

because,

Question 8.

Evaluate the integrals using substitution.

Answer:

Given: 

Let 2x = t ⇒ 2 dx = dt

When x = 1, t = 2 and when x = 2, t = 4

now, let 1/t = f(t)

then, f' (t) = -1/t²

because,

Question 9.

The value of the integral  is
A. 6

B. 0

C. 3

D. 4

Answer:

Given: 

let 

Now, let x = sin θ ⇒ dx = cos θ d θ

when, and when x = 1, θ = π/2

Now, let cot θ = t ⇒ - cosec² θ d θ

when,

= 6

Correct option is: (A)

Question 10.

If  then f'(x) is
A. 

B. 

C. 

D. 

Answer:

Given: 

Applying product rule,

⇒ 

= -x cos x + sin x – 0

⇒ f(x) = -x cos x + sinx

⇒ f'(x) = -[{x(-sinx )} + cosx ] + cosx

= x sin x – cos x + cos x

= x sin x

Correct answer is B.

---

Exercise 7.11

Question 1.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

let,

as,

Adding (1) and (2), we get

Question 2.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

let,

as,

Adding (1) and (2), we get

Question 3.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

Adding (1) and (2), we get

Question 4.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

Adding (1) and (2), we get

Question 5.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

As we can see that (x+2) ≤ 0 on [-5, -2] and (x+2) ≥ 0 on [-2,5]

Question 6.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

As we can see that (x-5)≤0 on [2,5] and (x+2)≥0 on [5,8]

⇒ I = 9

Question 7.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

Question 8.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

Question 9.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

Question 10.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

Adding (1) and (2), we get

Question 11.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

As we can see f(x) = sin²x and f(-x) = sin²(-x) = (sin (-x))² = (-sin x)² = sin²x.

i.e. f(x) = f(-x)

so, sin²x is an even function.

It is also known that if f(x) is an even function then,

Question 12.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

Adding (1) and (2), we get

⇒ I = π

Question 13.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

As we can see f(x) = sin⁷x and f(-x) = sin⁷(-x) = (sin (-x))⁷ = (-sin x)⁷ = -sin⁷x.

i.e. f(x) = -f(-x)

so, sin²x is an odd function.

It is also known that if f(x) is an odd function then,

Question 14.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

As we see, f(x)=cos⁵x and f(2π –x) = cos⁵(2π –x) = cos⁵x = f(x)

Question 15.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

Adding (1) and (2), we get

Question 16.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

Adding (1) and (2), we get

Here, if f(x) = log (sin x) and f(π – x)=log ( sin (π –x))= log (sin x ) = f(x)

Adding (1) and (2), we get

Let 2x = t ⇒ 2dx = dt

When x = 0, t = 0 and when x = π /2, t = π

Question 17.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

Adding (1) and (2), we get

Question 18.

By using the properties of definite integrals, evaluate the integrals

Answer:

Given: 

As we can see that (x-1)≤0 when 0≤x≤1 and (x-1)≥0 when 1≤x≤4

Question 19.

Show that  if f and g are defined as  and 

Answer:

Given: 

Adding (1) and (2), we get

Question 20.

The value of 
A. 0

B. 2

C. 

D. 1

Answer:

Given: 

It is also known that if f(x) is an even function then,

It is also known that if f(x) is an odd function then,

Correct answer is C

Question 21.

The value of 
A. 2

B. 

C. 0

D. -2

Answer:

Given: 

Adding (1) and (2), we get

Correct answer is (c)

---

Miscellaneous Exercise

Question 1.

Integrate the function: 

Answer:

Given: 

Let

Using partial differentiation:

let

⇒ 1 = A – Ax² + Bx – Bx² + Cx + Cx²

⇒ 1 = A + (B+C)x + (-A-B+C)x²

Equating the coefficients of x, x² and constant value. We get:

(a) A = 1

(b) B+C = 0 ⇒ B = -C

(c) -A –B +C =0

⇒ -1 – (-C) +C = 0

⇒ 2C = 1 ⇒ C = 1/2

So, B = -1/2

Put these values in equation (1)

Question 2.

Integrate the function: 

Answer:

Given: 

let

Multiply and divide by,

Question 3.

Integrate the function: 

Answer:

Given: 

let

put

Question 4.

Integrate the function: 

Answer:

Given: 

let 

Multiply and divide by x^-3, we get

Question 5.

Integrate the function: 

Answer:

Given: 

or we can write it as,

Let x = t⁶⇒ dx = 6t⁵dt

After division we get,

Question 6.

Integrate the function: 

Answer:

Given: 

Using partial differentiation:

let

⇒5x=A(x²+9)+(Bx+C)(x+1)

⇒ 5x = Ax² +9A+ Bx² +Bx+ Cx + C

⇒ 5x = 9A + C + (B+C)x + (A+B)x²

Equating the coefficients of x, x² and constant value. We get:

(a) 9A + C = 0 ⇒ C = -9A

(b) B+C = 5 ⇒ B = 5-C ⇒ B = 5-(-9A) ⇒ B = 5 + 9A

( c) A + B =0 ⇒ A = -B ⇒ A = -(5 + 9A) ⇒ 10A = -5 ⇒ A = -1/2

and C = 9/2 and B = 1/2

Put these values in equation (1)

Put x² = t ⇒ 2xdx = dt

Put the value in equ. (2)

Question 7.

Integrate the function: 

Answer:

Given: 

Let x – a = t ⇒ x = t + a ⇒ dx = dt

As,{ sin (A+B) = sin A cos B + cos A sin B}

Question 8.

Integrate the function: 

Answer:

Given: 

Question 9.

Integrate the function: 

Answer:

Given: 

Put sin x = t ⇒ cos x dx = dt

Question 10.

Integrate the function: 

Answer:

Given: 

Question 11.

Integrate the function: 

Answer:

Given: 

Multiply and divide by sin (a-b), we get

As,{ sin (A-B) = sin A cos B - cos A sin B}

Question 12.

Integrate the function: 

Answer:

Given: 

Now, let x⁴ = t ⇒ 4x³ dx = dt

And x³ dx = dt/4

Question 13.

Integrate the function: 

Answer:

Given: 

Let e^x = t ⇒ e^x dx = dt

Question 14.

Integrate the function: 

Answer:

Given: 

Using partial differentiation:

⇒1 = (Ax + B)(x² + 4)+(Cx + D)(x² + 1)

⇒ 1 = Ax³ +4Ax+ Bx² + 4B+ Cx³ + Cx + Dx² + D

⇒ 1 = (A+C)x³ +(B+D)x² +(4A+C)x + (4B+D)

Equating the coefficients of x, x², x³ and constant value. We get:

(a) A + C = 0 ⇒ C = -A

(b) B + D = 0 ⇒ B = -D

( c) 4A + C =0 ⇒ 4A = -C ⇒ 4A = A ⇒ 3A = 0 ⇒ A = 0 ⇒ C = 0

( d) 4B + D = 1 ⇒ 4B – B = 1 ⇒ B = 1/3 ⇒ D = -1/3

Put these values in equation (1)

Question 15.

Integrate the function: 

Answer:

Given: 

Let

Let cos x = t ⇒ -sin x dx = dt ⇒ sin x dx = dt

Question 16.

Integrate the function: 

Answer:

Given: 

Let

Let x⁴ = t ⇒ 4x³ dx = dt ⇒ x³ dx = dt/4

Question 17.

Integrate the function: 

Answer:

Given: f^’ (ax + b)[f(ax + b)]ⁿ

Let f(ax +b) = t ⇒ a .f^’ (ax + b)dx = dt

Question 18.

Integrate the function: 

Answer:

Given: 

As,{ sin (A+B) = sin A cos B + cos A sin B}

Question 19.

Integrate the function: 

Answer:

Given: 

let

As we know,

Now, first solve for I₁:

because,

Put it in equ. (2)

Question 20.

Integrate the function: 

Answer:

Given: 

let

Let x= cos²θ ⇒ dx = -2sinθ cosθ dθ

Question 21.

Integrate the function: 

Answer:

Given: 

Now let tan x = f(x)

⇒ f^’(x) = sec²x dx

Question 22.

Integrate the function: 

Answer:

Given: 

Let

Using partial differentiation:

let

⇒ x² + x + 1 = Ax² + 3Ax + 2A + Bx +2B + Cx² + 2Cx + C

⇒ x² + x + 1 = (2A+2B+C) + (3A+B+2C)x + (A+C)x²

Equating the coefficients of x, x² and constant value. We get:

(a) A + C = 1

(b) 3A + B + 2C = 1

( c) 2A+2B+C =1

After solving we get:

A=-2, B=1 and C=3

Question 23.

Integrate the function: 

Answer:

Given: 

Let x= cosθ ⇒ dx = -sin θ d θ

Question 24.

Integrate: 

Answer:

Given: 

Question 25.

Evaluate the definite integral:

Answer:

Given: 

Question 26.

Evaluate the definite integral:

Answer:

Given: 

Now let tan²x = t ⇒ 2 tan x sec²x dx = dt

And when x=0 then t=0 and when x=π /4 then t=1

Question 27.

Evaluate the definite integral:

Answer:

Given: 

First solve for I1:

Let 2 tan x = t ⇒ 2 sec²x dx dt

When x = 0 then t = 0 and when x = π /2 then t = ∞

Put this value in equ.(2)

Question 28.

Evaluate the definite integral: 

Answer:

Given: 

Now let sin x – cos x = t ⇒( cos x + sin x )dx = dt

i.e. f(x) = f(-x)

so, f(x) is an even function.

It is also known that if f(x) is an even function then,

Question 29.

Evaluate the definite integral:

Answer:

Given: 

Question 30.

Evaluate the definite integral: 

Answer:

Let 

Also, let sinx – cosx = t

Differentiating both sides, we get,

(cosx + sinx) dx = dt

When x = 0, t = -1

And when , t = 0

Now,

(sinx – cosx)² = t²

1 – 2 sinx.cosx = t²

Sin2x = 1 – t²

Putting all the values, we get the integral,

Question 31.

Evaluate the definite integral:

Answer:

Given: 

Let sin x = t ⇒ cos x dx = dt

When x =0 then t = 0 and when x = π /2 then t = 1

Question 32.

Evaluate the definite integral:

Answer:

Given: 

Adding (1) and (2), we get

Question 33.

Evaluate the definite integral:

Answer:

Given: 

First solve for I1:

As we can see that (x-1)≥0 when 1≤x≤4

Now solve for I2:

As we can see that (x-2)≤0 when 1≤x≤2 and (x-2)≥0 when 2≤x≤4

Now solve for I3:

As we can see that (x-3)≤0 when 1≤x≤3 and (x-3)≥0 when 3≤x≤4

Question 34.

Prove: 

Answer:

Given: 

Using partial differentiation:

⇒ 1 = Ax² +Ax+ B+Bx+ Cx²

⇒ 1 = B + (A+B)x + (A+C)x²

Equating the coefficients of x, x² and constant value. We get:

(a) B = 1

(b) A + B = 0 ⇒ A = -B ⇒ A = -1

( c) A + C =0 ⇒ C = -A ⇒ C = 1

Put these values in equation (1)

⇒ L.H.S = R.H.S

Hence proved.

Question 35.

Prove: 

Answer:

Given: 

L.H.S = R.H.S

Hence Proved.

Question 36.

Prove: 

Answer:

Given: 

As we can see f(x) =x¹⁷ .cos⁴x and f(-x) = (-x)¹⁷ .cos⁴(-x) = -x¹⁷ .cos⁴x

i.e. f(x) = -f(-x)

so, it is an odd function.

It is also known that if f(x) is an odd function then,

Hence proved.

Question 37.

Prove: 

Answer:

Given: 

First solve for I1:

Let cos x = t ⇒ -sin x dx = dt ⇒ sinx dx = -dt

When x=0 then t= 1 and when x = π /2 then t = 0

Put in equ. (2)

L.H.S = R.H.S

Hence Proved.

Question 38.

Prove: 

Answer:

Given: 

First solve for I1:

Let tan x = t ⇒ sec² x dx = dt

When x=0 then t= 0 and when x = π /2 then t = 1

Put in equ. (2)

L.H.S = R.H.S

Hence Proved.

Question 39.

Prove: 

Answer:

Given: 

First solve for I1:

Let 1 - x² = t ⇒ -2 x dx = dt

When x=0 then t= 1 and when x = 1 then t = 0

Put in equ. (2)

L.H.S = R.H.S

Hence Proved.

Question 40.

Evaluate  as a limit of a sum.

Answer:

Given: 

Here, a=0, b=1, and f(x)=e^2-3x and h = 1/n

Question 41.

Choose the correct answers

 is equal to
A. 

B. 

C. 

D. 

Answer:

Given: 

Put e^x= t ⇒ e^x dx = dt

Hence, correct option is (A).

Question 42.

Choose the correct answers

 is equal to
A. 

B. 

C. 

D. 

Answer:

Given: 

Put sin x + cos x= t ⇒ cos x – sin x = dt

Hence, correct option is (B).

Question 43.

Choose the correct answers

If f (a + b – x) = f (x), then  is equal to
A. 

B. 

C. 

D. 

Answer:

Given: 

Hence, correct option is (D).

Question 44.

Choose the correct answers

The value of 
A. 1

B. 0

C. –1

D. 

Answer:

Given: 

Adding (1) and (2), we get

Hence, correct option is (B).

---

Exercise 7.2

Question 1.

Integrate the functions.

Answer:

Let 1+x² = t

⇒ 2x dx = dt

Now, 

= log|t| + C

= log|1+x²| + C

= log(1+x²) + C

Question 2.

Integrate the functions.

Answer:

Let log|x| = t

Now, 

⇒ 

⇒ 

Question 3.

Integrate the functions.

Answer:

let 1+logx =t

= log|t| + C

= log | 1+ logx| + C

Question 4.

Integrate the functions.

sin x sin (cos x)

Answer:

Let cosx = t

⇒ -sinxdx = dt

⇒ 

=-[-cost] + C

= cost + C

= cos(cosx) + C

Question 5.

Integrate the functions.
sin (ax + b) cos (ax + b)

Answer:

Let I = ∫sin (ax + b) cos (ax + b) dx

We know that,

sin 2A = 2sinA.cosA

Therefore,

sin (ax + b) cos (ax + b) = 

Let 2(ax+b) = t

⇒ 2adx = dt

⇒ 

=

Question 6.

Integrate the functions.

Answer:

Let ax + b =t

⇒ adx = dt

Question 7.

Integrate the functions.

Answer:

Let (x +2) = t

⇒ dx = dt

Question 8.

Integrate the functions.

Answer:

Let 1 +2x² =t

⇒ 4xdx = dt

⇒ 

⇒ 

⇒ 

⇒ 

Question 9.

Integrate the functions.

Answer:

Let x² + x+ 1= t

Differentiating both sides, we get,

⇒ (2x + 1)dx = t dt

Therefore,

⇒ 

⇒ 

⇒ 

⇒ 

Question 10.

Integrate the functions.

Answer:

Now, Let 

⇒ 

⇒ 

= 2log|t| + C

= 2log|| + C

Question 11.

Integrate the functions.

Answer:

Let x+ 4 = t

⇒ dx = dt

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Question 12.

Integrate the functions.

Answer:

Let x³ -1 = t

⇒ 3x²dx = dt

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Question 13.

Integrate the functions.

Answer:

Let 2 +3x³ = 1

⇒ 9x² dx = dt

⇒ 

⇒ 

⇒ 

⇒ 

Question 14.

Integrate the functions.

Answer:

Let log x = t

⇒ 

⇒ 

⇒ 

⇒ 

Question 15.

Integrate the functions.

Answer:

Let 9 – 4x² = t

⇒ -8xdx = dt

⇒ 

⇒ 

⇒ 

Question 16.

Integrate the functions.

Answer:

Let 2x +3 =t

⇒ 2dx = dt

⇒ 

⇒ 

⇒ 

Question 17.

Integrate the functions.

Answer:

Let x² = t

⇒ 2xdx = dt

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Question 18.

Integrate the functions.

Answer:

let tan-1 x = t

⇒ 

⇒ 

= e^t + C

= 

Question 19.

Integrate the functions.

Answer:

We have, 

Dividing numerator and denominator by e^x, we get,

Let e^x + e^-x = t
Differentiating both sides, we get,

(e^x - e^-x )dx = dt

Now the integral becomes,

= log |t| +C

= log|e^x + e^-x| + C

Question 20.

Integrate the functions.

Answer:

Let 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

= log |t| +C

= log|| + C

Question 21.

Integrate the functions.

tan² (2x – 3)

Answer:

tan² (2x – 3) = sec² (2x – 3) – 1

Let 2x -3 = t

⇒ 2dx = dt

⇒ 

⇒ 

⇒ 

⇒ 1/2 tan t - t + C

⇒ 1/2 tan (2x - 3) - (2x - 3) + C

Question 22.

Integrate the functions.

sec² (7 – 4x)

Answer:

Let 7 – 4x = t

⇒ -4dx = dt

⇒ 

⇒ 

⇒ 

Question 23.

Integrate the functions.

Answer:

let sin^-1 x =t

⇒ 

⇒ 

⇒ 

⇒ 

Question 24.

Integrate the functions.

Answer:

let 3cosx + 2sinx = t

(-3sinx + 2cosx)dx = dt

⇒ 

⇒ 

⇒ 

⇒ 

Question 25.

Integrate the functions.

Answer:

Let (1 – tanx) = t

⇒ -sec²xdx = dt

⇒ 

⇒ 

⇒ 

Question 26.

Integrate the functions.

Answer:

Let 

⇒ 

⇒ 

= 2sint + C

= 2sin + C

Question 27.

Integrate the functions.

Answer:

Let sin2x = t

⇒ 2cos2xdx = dt

.

⇒ 

⇒ 

Question 28.

Integrate the functions.

Answer:

Let 1 + sinx = t

⇒ cosx dx = dt

⇒ 

⇒ 

⇒ 

⇒ 

Question 29.

Integrate the functions.

cot x log sin x

Answer:

Let Log sinx = t

⇒ 

⇒ cotx dx = dt

⇒ 

⇒ 

⇒ 

Question 30.

Integrate the functions.

Answer:

Let 1+ cosx = t

⇒ -sinx dx = dt

⇒ 

= - log|t| + C

= -log|1 + cosx| + C

Question 31.

Integrate the functions.

Answer:

Let 1 + cosx = t

⇒ -sinx dx = dt

⇒ 

⇒ 

⇒ 

⇒ 

Question 32.

Integrate the functions.

Answer:

Let I = 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Let sinx + cosx = t

⇒ (cosx-sinx)dx = dt

Therefore, I = 

⇒ 

⇒ 

Question 33.

Integrate the functions.

Answer:

Let I =

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Let cosx - sinx = t

⇒ (-sinx-cosx)dx = dt

Therefore, I = 

⇒ 

⇒ 

Question 34.

Integrate the functions.

Answer:

Let I = 

⇒ 

⇒ 

⇒ 

Let tanx = t

⇒ sec²x dx = dt

⇒ I = 

= 2+C

= 2 + C

Question 35.

Integrate the functions.

Answer:

let 1 + log x = t

⇒  = dt

⇒ 

⇒ 

⇒ 

Question 36.

Integrate the functions.

Answer:

Let (x+logx) = t

⇒ 

⇒ 

⇒ 

⇒ 

Question 37.

Integrate the functions.

Answer:

Let x⁴ = t

⇒ 4 x³ dx = dt

⇒ 

⇒ 

Let tan-1 = v

⇒ 

Thus, we get,

⇒ 

⇒ 

⇒ 

⇒ 

Question 38.

Choose the correct answer:


A. 10^x – x¹⁰ + C 
B. 10^x + x¹⁰ + C
C. (10^x – x¹⁰)–1 + C 
D. log (10^x + x¹⁰) + C

Answer:

Let x¹⁰ + 10^x = t

⇒ (10x⁹ + 10^x log_e10)dx = dt

⇒ 

= log t + C

= log(x¹⁰ + 10^x) + C

Question 39.

 equals
A. tan x + cot x + C B. tan x – cot x + C

C. tan x cot x + C D. tan x – cot 2x + C

Answer:

Let I = 

⇒ 

⇒ 

⇒ 

= tanx – cotx + C

---

Exercise 7.3

Question 1.

Find the integrals of the functions.

Answer:

sin²(2x+5) 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Question 2.

Find the integrals of the functions.

Answer:

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Question 3.

Find the integrals of the functions.

Answer:

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Question 4.

Find the integrals of the functions.

Answer:

Let I = sin³(2x+1)

⇒ 

⇒ 

Let cos (2x+1) = t

=> -2sin(2x+1)dx = dt

=> sin(2x+1)dx = 

⇒ 

⇒ 

⇒ 

⇒ 

Question 5.

Find the integrals of the functions.

Answer:

Let I = 

⇒ 

⇒ 

Let cosx = t

⇒ -sinx.dx = dt

⇒ I 

⇒ 

⇒ 

⇒ 

⇒ 

Question 6.

Find the integrals of the functions.

Answer:

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Question 7.

Find the integrals of the functions.

Answer:

⇒ 

⇒ 

⇒ 

Question 8.

Find the integrals of the functions.

Answer:

⇒ 

⇒ 

⇒ 

Question 9.

Find the integrals of the functions.

Answer:

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Question 10.

Find the integrals of the functions.

Answer:

sin⁴x = sin²xsin²x

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Now, 

⇒ 

⇒ 

⇒ 

Question 11.

Find the integrals of the functions.

Answer:

cos⁴2x = (cos²2)²

⇒ 

⇒ 

⇒

⇒ 

⇒ 

⇒ 

Now, 

⇒ 

Question 12.

Find the integrals of the functions.

Answer:

⇒ 

⇒ 

= 1- cosx

⇒ 

= x – sinx + C

Question 13.

Find the integrals of the functions.

Answer:

Using the identity , we have

Now, using the identity sin 2x = 2 sin x cos x, we have

Using identity 2 cos A cos B = cos (A + B) + cos (A - B), we have

= 2[cos(x) +cosα]

= 2cosx + 2 cosα

⇒ 

= 2[sinx + xcosα] + C

Question 14.

Find the integrals of the functions.

Answer:

⇒ 

⇒ 

Let sinx + cosx = t

⇒ (cosx-sinx)dx =dt

⇒ 

⇒ 

= -t^-1 + C

⇒ 

⇒ 

Question 15.

Find the integrals of the functions.

Answer:

tan³2xsec2x = tan²2xtan2xsec2x

= (sec²2x -1)tan2xsec2x

= sec²2x.tan2xsec2x-tan2xsec2x

⇒ 

⇒ 

Now, Let sec2x = t

⇒ 2sec2xtan2x dx = dt

Thus, 

⇒ 

⇒ 

Question 16.

Find the integrals of the functions.

Answer:

tan⁴x = tan²x.tan²x

= (sec²x-1) tan²x

= sec²x tan²x- tan²x

= sec²x tan²x- (sec²x-1)

= sec²x tan²x- sec²x+1

Now, 

⇒ 

Now, let tanx = t

=> sec²x dx =dt

⇒ 

⇒ 

Question 17.

Find the integrals of the functions.

Answer:

⇒ 

⇒ 

= tanxsecx + cotxcosecx

⇒ Now, 

= secx – cosecx + C

Question 18.

Find the integrals of the functions.

Answer:

⇒ 

= sec²x

Now, 

= tanx + C

Question 19.

Find the integrals of the functions.

Answer:

⇒ 

⇒ 

Now, 

let tanx = t

⇒ sec²x dx = dt

⇒ 

⇒ 

⇒ 

Question 20.

Find the integrals of the functions.

Answer:

Now, 

Let 1 + sin2x = t

⇒ 2cos2x dx = dt

Thus, 

⇒ 

⇒ 

⇒ 

= log|sinx + cosx| + C

Question 21.

Find the integrals of the functions.

Answer:

sin^-1(cosx)

Let cosx = t.......(1)

Then, sinx = 

Differentiating both sides of (1), we get,

⇒ (-sinx)dx = dt

⇒ dx = 

⇒ dx = 

⇒ Now, 

⇒ 

Let sin^-1 t = v

⇒ 

⇒ ∫sin^-1(cosx) dx = - ∫vdv

⇒ 

⇒ 

 …(1)

We know that,

sin-1x + cos-1 x = 

⇒ sin^-1(cosx)= 

Now, substituting in eq(1), we get,

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Question 22.

Find the integrals of the functions.

Answer:

⇒ 

⇒ 

⇒ 

Now, 

⇒ 

⇒ 

Question 23.

A.  B. 

C.  D. 

Answer:

We know that,
∫sec²x dx = tan x + c
∫cosec²x dx = - cot x + c
= tanx + cotx + C.
A: Answer

Question 24.

A.  B. 

C.  D. 

Answer:

Let x.e^x = t
Differentiating both sides we get,

⇒ (e^x.x + e^x.1)dx = dt

⇒ e^x(x + 1) = dt

Now, 

= tan t + C

= tan(e^x.x) + C

---

Exercise 7.4

Question 1.

Integrate the functions.

Answer:

Let x³ = t

⇒ 3x² dx = dt

⇒ 

= tan^-1t + C

= tan^-1(x³) + C

Question 2.

Integrate the functions.

Answer:

Let 2x= t

⇒ 2dx = dt

⇒ 

⇒ 

Question 3.

Integrate the functions.

Answer:

Let 2 – x = t

⇒ -dx = dt

⇒ 

⇒ 

⇒ 

Question 4.

Integrate the functions.

Answer:

Let 5x = t

⇒ 5dx = dt

⇒ 

⇒ 

⇒ 

Question 5.

Integrate the functions.

Answer:

Let = t

⇒ 2dx = dt

⇒ 

⇒ 

⇒ 

Question 6.

Integrate the functions.

Answer:

Let x³ = t

⇒ 3x² dx = dt

⇒ 

⇒ 

⇒ 

Question 7.

Integrate the functions.

Answer:

For ,

Let x² -1 = t

⇒ 2x dx = dt

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Question 8.

Integrate the functions.

Answer:

Let x³ = t

⇒ 3x² dx = dt

⇒ 

⇒ 

⇒ 

Question 9.

Integrate the functions.

Answer:

Let tanx = t

⇒ sec²xdx = dt

⇒ 

⇒ 

⇒ 

Question 10.

Integrate the functions.

Answer:

Let x+1 = t

⇒ dx = dt

⇒ 

⇒ 

⇒ 

⇒ 

Question 11.

Integrate the functions.

Answer:

⇒ 

Let 3x+1= t

⇒ 3dx = dt

⇒ 

⇒ 

⇒ 

Question 12.

Integrate the functions.

Answer:

Let x+3= t

⇒ dx = dt

⇒ 

⇒ 

⇒ 

Question 13.

Integrate the functions.

Answer:

Let 

⇒ dx = dt

⇒ 

⇒ 

⇒ 

Question 14.

Integrate the functions.

Answer:

Let 

⇒ dx = dt

⇒ 

⇒ 

⇒ 

Question 15.

Integrate the functions.

Answer:

(x - 1)(x - b) can be written as x² – (a+b)x + ab.

Then, x² – (a+b)x + ab = x² – (a+b)x + 

⇒ 

⇒ 

Let x = 

⇒ dx = dt

⇒ 

⇒ 

⇒ 

Question 16.

Integrate the functions.

Answer:

Let 4x + 1 = 

⇒ 4x + 1 = A(4x + 1) + B

⇒ 4x + 1 = 4Ax+ A+ B

Now, equating the coefficients of x and constant term on both sides, we get,

4A = 4

⇒ A = 1

A + B = 1

⇒ B = 0

Let 2x² + x – 3 =t

⇒ (4x + 1) dx = dt

⇒ 

⇒ 

 2

Question 17.

Integrate the functions.

Answer:

Let x + 2 = 

⇒ x + 2 = A(2x) + B

Now, equating the coefficients of x and constant term on both sides, we get,

2A = 1

⇒ A = 

B = 2

⇒ x + 2 =  (2x) + 2

⇒ 

⇒ 

Now, 

Let x² - 1 = t

⇒ (2x)dx = dt

⇒ 

⇒ 

And 

⇒ 

Question 18.

Integrate the functions.

Answer:

Let 5x - 2 = 

⇒ 5x - 2 = A(2 + 6x) + B

Now, equating the coefficients of x and constant term on both sides, we get,

6A = 5

⇒ A = 

2A + B = -2

⇒ B = 

⇒ 5x - 2 =  (2 + 6x) + 

⇒ 

⇒ 

Now, 

Let 1+2x+3x² = t

⇒ (2 + 6x)dx = dt

⇒ 

= log|1+2x+3x²| …(1)

And 

1+2x+3x² = 

⇒ 

⇒ 

⇒ 

⇒ 

 …(2)

Thus, from (1) and (2), we get,

⇒ 

⇒ 

Question 19.

Integrate the functions.

Answer:

Let 6x + 7 = 

⇒ 6x + 7 = A(2x - 9) + B

Now, equating the coefficients of x and constant term on both sides, we get,

2A = 6

⇒ A = 3

-9A + B = 7

⇒ B = 34

⇒ 6x + 7 = 3 (2x - 9) + 34

⇒ 

⇒ 

Now, 

Let x² – 9x + 20 = t

⇒ (2x - 9)dx = dt

 ----------(1)

And

x² – 9x + 20 = 

⇒ 

⇒ 

 …(2)

Thus, from (1) and (2), we get,

⇒ 

⇒ 

Question 20.

Integrate the functions.

Answer:

Let x + 2 = 

⇒ x + 2 = A(4 -2x) + B

Now, equating the coefficients of x and constant term on both sides, we get,

-2A = 1

⇒ A = 

4A + B = 2

⇒ B = 4

⇒ x + 2 = 

Now, 

⇒ 

Now, let us consider, 

Let 4x – x² = t

⇒ (4 -2x) dx= dt

 …(1)

And, Now let us consider, 

Then, 4x – x² = -(-4x + x²)

= (-4x + x² + 4 – 4)

= 4 – (x - 2)²

= (2)² – (x - 2)²

 …(2)

using eq. (1) and (2), we get,

⇒ 

Question 21.

Integrate the functions.

Answer:

⇒ 

⇒ 

⇒ 

Now, Let us consider 

Let x² + 2x + 3 = t

⇒ (2x + 2)dx = dt

 … (1)

And, now let us consider 

⇒ x² + 2x + 3 = x² + 2x + 1 + 2 = (x +1)² + (²

⇒ 

Using eq. (1) and (2), we get,

⇒ 

⇒ 

Question 22.

Integrate the functions.

Answer:

Let x + 3 = 

⇒ x + 3 = A(2x-2) + B

Now, equating the coefficients of x and constant term on both sides, we get,

2A = 1

⇒ A = 

-2A + B = 3

⇒ B = 4

⇒ x + 3 = 

Now, 

⇒ 

Now, Let us consider 

Let x2 – 2x – 5 = t

⇒ (2x -2)dx = dt

And, now let us consider, 

⇒ 

⇒ 

 …(2)

Using eq. (1) and (2), we get,

⇒ 

⇒ 

Question 23.

Integrate the functions.

Answer:

Let 5x + 3 = 

⇒ 5x + 3 = A(2x + 4) + B

Now, equating the coefficients of x and constant term on both sides, we get,

2A = 5

⇒ A = 

4A + B = 3

⇒ B = -7

⇒ 5x + 3 = 

Now, 

⇒ 

Now, let us consider, 

Let x² +4x + 10= t

⇒ (2x + 4) dx= dt

 … (1)

And, Now let us consider, 

⇒ 

⇒ 

 …(2)

using eq. (1) and (2), we get,

⇒ 

⇒ 

Question 24.

Choose the correct answer:


A. 

B. 

C. 

D. 

Answer:

⇒ 

⇒ 

Question 25.

Choose the correct answer: 


A. 

B. 

C. 

D. 

Answer:

⇒ 

⇒ 

⇒ 

⇒ 

---

Exercise 7.5

Question 1.

Integrate the rational functions.

Answer:

Let 

⇒ x = A(x + 2) + B(x +1)

On comparing the coefficients of x and constant term, we get,

A + B = 1

2A + B = 0

On solving above two equations, we get,

A = -1 and B = 2

Thus,

= -log|X + 1| + 2log|x + 2| + C

= log (x+2)² – log |x + 1| + C

Question 2.

Integrate the rational functions.

Answer:

Let 

⇒ 1= A(x - 3) + B(x + 3)

On comparing the coefficients of x and constant term, we get,

A + B = 0

-3A + 3B = 1

On solving above two equations, we get,

A =  and B = 

Thus,

Question 3.

Integrate the rational functions.

Answer:

Let 

⇒ 3x -1 = A(x-2)(x-3) + B(x-1)(x-3) + C(x-1)(x-2) …(1)

Substituting x = 1, 2 and 3 respectively in equation (1), we get,

A = 1, B = -5 and C = 4

Thus,

= log|x -1| -5log|x-2| + 4log|x-3| + C

Question 4.

Integrate the rational functions.

Answer:

Let 

⇒ x = A(x-2)(x-3) + B(x-1)(x-3) + C(x-1)(x-2) …(1)

Substituting x = 1, 2 and 3 respectively in equation (1), we get,

A = , B = -2 and C = 

Thus,

Question 5.

Integrate the rational functions.

Answer:

Let 

⇒ 2x = A(x + 2)+ B(x + 1) …(1)

Substituting x = -1 and -2 respectively in equation (1), we get,

A = -2, B = 4

Thus,

= 4log|x + 2| -2log|x + 1| + C

Question 6.

Integrate the rational functions.

Answer:

On dividing 1 – x² by x(1 - 2x), we get,

 …(1)

Now, let 

(2 – x) = A(1 – 2x) + Bx …(2)

Now, substituting x = 0 and  in equation (2), we get,

A = 2 and B = 3

Thus, 

Now, putting this value in equation (2), we get,

Question 7.

Integrate the rational functions.

Answer:

Let 

x = (Ax + B)(x -1) + C (x2 + 1)

⇒ x = Ax² – Ax + Bx – B + Cx² + C

Equating the coefficients of x2, x and constant term, we get,

A + C = 0

-A + B = 0

-B + c = 0

On solving these equation, we get,

A = , B =  and C = 

Thus,

Now, let us consider, ,

Let (x² + 1) = t

2xdx = dt

Thus,

Therefore,

Question 8.

Integrate the rational functions.

Answer:

Let 

⇒ x = A(x - 1)(x + 2) + B(x + 2) + C( x – 1)² …(1)

Substituting x = -1 in equation (1), we get,

B = 

Equating the coefficients of x2 and constant term, we get,

A + C = 0

-2A + 2B + C = 0

A =  and c = 

Thus,

Question 9.

Integrate the rational functions.

Answer:

Let 

⇒3x + 5 = A(x - 1)(x + 1) + B(x + 1) + C( x – 1)²

⇒3x + 5 = A(x² - 1) + B(x + 1) + C( x² + 1 – 2x) …(1)

Substituting x = 1 in equation (1), we get,

B = 4

Equating the coefficients of x² and x, we get,

A + C = 0

b – 2C = 3

A =  and c = 

Thus,

Question 10.

Integrate the rational functions.

Answer:

Let 

⇒2x – 3 = A(x - 1)(2x + 3) + B(x + 1)(2x + 3) + C(x+1)( x – 1)

⇒ 2x – 3 = A(2x² + x – 3) + B(2x² + 5x + 3) + C(x² - 1)

⇒2x - 3 = (2A + 2B + C)x² + (A + 5B)x + (-3A + 3B – C)

Equating the coefficients of x² and x, we get,

B =, A =  and C = 

Thus,

Question 11.

Integrate the rational functions.

Answer:

Let 

⇒ 5x = A(x + 2)(x - 2) + B(x + 1)(x - 2) + C(x+1)( x + 2) …(1)

Substituting x = -1, -2 and 2 respectively in equation (1), we get,

A =, B =, and C = 

Thus,

Question 12.

Integrate the rational functions.

Answer:

Dividing (x³ + x + 1) by x² -1, we get,

Let 

Now, 2x + 1 = A(x – 1) + B(x + 1) …(1)

Substituting x = 1 and -1 in equation (1), we get,

A =  and B = 

Thus,

Question 13.

Integrate the rational functions.

Answer:

Let 

⇒ 2 = A(1 – x²) + (Bx + C)(1 – x)

⇒ 2 = A + Ax2 + Bx – Bx2 + C – Cx

On comparing the coefficients of x², x and constant term, we get,

A – B = 0

B – C = 0

A + C = 0

On solving these equations, we get,

A = 1, B =1 and C = 1

Thus,

Question 14.

Integrate the rational functions.

Answer:

Let 

⇒ 3x -1 = A(x + 2) + B

Equating the coefficients of x and constant term, we get,

A = 3

2A + B = -1

B = -7

Thus,

Question 15.

Integrate the rational functions.

Answer:

Let 

1 = A(x-1)(x²+1) + B(x+1)(x²+1) + (Cx + D)(x² - 1)

1 = A(x³ + x – x² -1) + B(x³ + x + x² + 1) + Cx³ + Dx² - Cx - D

1 = (A + B + C)x³ + (-A + B + D)x² + (A + B - C)x + (-A + B - D)

Equating the coefficients of x³, x², x and constant term, we get,

(A + B + C) = 0

(-A + B + C) = 0

(A + B - C) = 0

(-A + B - D) = 0

On solving these equations, we get,

A = 

Therefore,

Question 16.

Integrate the rational functions.

 [Hint: multiply numerator and denominator by x^n-1 and put xⁿ=t]

Answer:

Multiplying numerator and denominator by x^n-1, we get,

Let xⁿ = t

x^n-1dx = dt

Therefore,

Let 

1 = A(1+t) + Bt ...(1)

Substituting t = 0, -1 in equation (1), we get,

A =1 and B = -1

Thus,

Question 17.

Integrate the rational functions.

Answer:

Let sinx = t

cosx dx = dt

Therefore,

Let 

1 = A(2 – t) + B (1 – t) …(1)

Substituting t = 2 and then t = 1 in equation (1), we get,

Therefore,

Question 18.

Integrate the rational functions.

Answer:

Now, 

Let 

4x² + 10 = (Ax + B)(x² + 4) + (Cx + D)(x² + 3)

⇒ 4x² + 10 = Ax³ + 4Ax + Bx² + 4B + Cx³ + 3Cx + Dx² + 3D

⇒ 4x² + 10 = (A+C)x³ + (B + D)x² + (4A + 3C)x + (4B + 3D)

Equating the coefficients of x³, x², x and constant term, we get,

A + C = 0

B + D = 0

4A + 3C = 0

4B + 3 B = 0

On solving these equations, we get,

A = 0, B = -2, C = 0 and D = 6

Therefore,

Question 19.

Integrate the rational functions.

Answer:

Now, 

Now, let x² = t

2xdx = dt

Thus,  …(1)

Let 

1 = A(t + 3) + B(t + 1)……………….(2)

Substituting t = -3 and -1 in (2), we get

A = and B = 

Question 20.

Integrate the rational functions.

Answer:

Now, 

Multiplying numerator and denominator by x³, we get,

Now, let x⁴ = t

4x³dx = dt

Thus, 

Let 

1 = A(t – 1) + Bt …(1)

Substituting t = 0 and 1 in (1), we get

A = -1 and B = 1

Question 21.

Integrate the rational functions.

 [Hint: Put e^x = t]

Answer:

Now, 

Let e^x = t

e^x dx = dt

Let 

1 = A(t – 1) + Bt …(1)

Substituting t =1 and t = 0 in equation (1), we get,

A = -1 and B = 1

Question 22.

Choose the correct answer


A. 
B. 

C. 

D. 

Answer:

Let 

x = A(x – 2) + B(x - 1) …(1)

Substituting x =1 and 2 in (1), we get,

A = -1 and B = 2

= -log|x – 1| + 2log| x – 2| + C

Question 23.

Integrate the rational functions.


A. 

B. 

C. 

D. 

Answer:

Let 

1 = A(X² + 1) + (Bx + C)

Equating the coefficients of x2, x and constant term, we get,

A + B = 0

C = 0

A = 1

On solving these equations, we get,

A = 1, B = -1 and C = 0

---

Exercise 7.6

Question 1.

Integrate the functions.

Answer:

Let I = xsinx

Now, integrating by parts, we get,

Question 2.

Integrate the functions.

Answer:

Let I = sin3x

Now, integrating by parts, we get,

Question 3.

Integrate the functions.

Answer:

Let I = x²e^x

Now, integrating by parts, we get,

Again integrating by parts, we get,

Question 4.

Integrate the functions.

Answer:

Let I = xlogx

Now, integrating by parts, we get,

Taking, Logarithmic function as first function and algebraic function as second function,

Question 5.

Integrate the functions.

Answer:

Let I = xlog2x

Now, integrating by parts, we get,

Question 6.

Integrate the functions.

Answer:

Let I = x²logx

Now, integrating by parts, we get,

Question 7.

Integrate the functions.

Answer:

Let I = xsin^-1x

Now, integrating by parts, we get,

Question 8.

Integrate the functions.

Answer:

Let I = xtan^-1x

Now, integrating by parts, we get,

Question 9.

Integrate the functions.

Answer:

Let I = xcos^-1x

Now, integrating by parts, we get,

 …(1)

Now, 

Now, substituting in (1), we get,

Question 10.

Integrate the functions.

Answer:

Let I = (sin^-1x)²

Now, integrating by parts, we get,

Question 11.

Integrate the functions.

Answer:

Let 

Now, integrating by parts, we get,

Question 12.

Integrate the functions.

Answer:

Let I = xsec²x

Now, integrating by parts, we get,

= xtanx + log|cosx| + C

Question 13.

Integrate the functions.

Answer:

Let 

So, now integrating by parts, we get,

Question 14.

Integrate the functions.

Answer:

Let 

Integrating by parts, we get,

Question 15.

Integrate the functions.

Answer:

Now, Let I = I₁ + I₂ …(1)

Where, 

So, now

Integrating by parts, we get,

 …(2)

Now, 

Integrating by parts, we get,

= xlogx –x + C₂ …(3)

Now putting the value of I₁ and I₂, in (1), we get,

I 

Question 16.

Integrate the functions.

Answer:

Now,

Let 

We know that,

Thus,

Question 17.

Integrate the functions.

Answer:

Now,

Let 

We know that,

Thus,

Question 18.

Integrate the functions.

Answer:

Now,

Let 

We know that,

Thus,

Question 19.

Integrate the functions.

Answer:

Let I 

Now, let

We know that,

Thus,

Question 20.

Integrate the functions.

Answer:

Now, let f(x) = 

We know that,

Thus,

Question 21.

Integrate the functions.

Answer:

Let I = e^2xsinx

Integrating by parts, we get,

I 

Again, integrating by parts, we get,

Question 22.

Integrate the functions.

Answer:

Let x = tanƟ

⇒ dx = sec²ƟdƟ

Thus,

Integrating by parts, we get

= 2[ƟtanƟ + log|cosƟ| + C

Question 23.

Choose the correct answer:


A. 

B. 

C. 

D. 

Answer:

Let I = 

Also, let x³ = t

⇒ 3x²dx = dt

Thus,

⇒ I = 

Question 24.

Choose the correct answer:

A. 

B. 

C. 

D. 

Answer:

Let I = 

Also, let secx = f(x)

⇒ secxtanx = f’(x)

We know that, 

Thus, I = e^xsecx + C

---

Exercise 7.7

Question 1.

Integrate :

Answer:

We know that,

⇒ 

Therefore,

⇒ 

⇒ 

Question 2.

Integrate :

Answer:

⇒ 

Let 2= t

_⇒

We know that,

⇒ 

Therefore,

⇒ 

⇒ 

⇒ 

⇒ 

Question 3.

Integrate :

Answer:

⇒ 

⇒ 

⇒ 

⇒ We know that,

⇒ 

Therefore,

⇒ 

⇒ 

Question 4.

Integrate : 

Answer:

⇒ 

⇒ 

⇒ 

We know that,

Therefore,

⇒ 

Question 5.

Integrate : 

Answer:

⇒ 

⇒ 

⇒ 

⇒ 

We know that,

⇒ 

⇒ 

Question 6.

Integrate : 

Answer:

⇒ 

⇒ 

We know that,

⇒ 

⇒ 

Question 7.

Integrate :

Answer:

⇒ 

⇒ 

⇒ 

⇒ 

We know that,

⇒ 

Therefore,

⇒ 

⇒ 

Question 8.

Integrate : 

Answer:

⇒ 

⇒ 

⇒ 

We know that,

⇒ 

Therefore,

⇒ 

⇒ 

Question 9.

Integrate :

Answer:

⇒ 

⇒ 

⇒ 

We know that,

⇒ 

Therefore,

⇒ 

Question 10.

Choose the correct answer

 is equal to
A. 

B. 

C. 

D. 

Answer:

We know that,

⇒ 

Therefore,

⇒ 

Question 11.

Choose the correct answer

 is equal to

A. 

B. 

C. 

D. 

Answer:

⇒ 

⇒ 

⇒ 

We know that,

⇒ 

Therefore,

⇒ 

---

Exercise 7.8

Question 1.

Evaluate using limit of sums 

Answer:

f(x) is continuous in [a,b]

here h=b-a/n

Question 2.

Evaluate using limit of sums 

Answer:

f(x) is continuous in [0,5]

here h=5/n

Question 3.

Evaluate using limit of sums 

Answer:

f(x) is continuous in [2,3]

here h=1/n

Question 4.

Evaluate using limit of sums 

Answer:

f(x) is continuous in [1,4]

here h=3/n

Question 5.

Evaluate using limit of sums 

Answer:

f(x) is continuous in [-1,1]

here h=2/n

Which is g.p with common ratio e^1/n.

Whose sum is 

=-1

As h=2/n

=e-e^-1

Question 6.

Evaluate using limit of sums 

Answer:

F(x)=h(x)+g(x)

Solving for h(x)

. h(x) is continuous in [0,4]

here h=4/n

=8

Now solving for g(x)

g(x) is continuous in [0,4]

here h=4/n

Which is g.p with common ratio e^1/n

Whose sum is 

As h=4/n

=(e⁸-1)

Now for f(x)=h(x)+g(x)

=8+ e⁸-1

---

Exercise 7.9

Question 1.

Evaluate 

Answer:

Let I = 

⇒ I = 2

∴ 

Question 2.

Evaluate 

Answer:

Let I = 

[]

⇒ I = log |3| - log |2|

⇒ I = log 3/2

Question 3.

Evaluate 

Answer:

Let I = 

[]

Question 4.

Evaluate 

Answer:

Let I = 

⇒ I = 

 [∫sinx dx=-cosx]

⇒ I = - (cos 2×π/4 – cos 0)/2

⇒ I = - (cos π/2 – cos 0)/2 = - (0 – 1)/2

⇒ I = 1/2

∴  = 1/2

Question 5.

Evaluate 

Answer:

Let I = 

⇒ I = 

[]

⇒ I = � × (0 – 0) = 0

∴  = 0

Question 6.

Evaluate 

Answer:

Let I = 

⇒ I = 

⇒ I =  = e⁵ – e⁴[]

⇒ I = e⁴ (e – 1)

∴  = e⁴ (e – 1)

Question 7.

Evaluate 

Answer:

Let I = 

⇒ I = []

⇒ I =  = 

⇒ I = 

⇒ I = 

∴  = 

Question 8.

Evaluate 

Answer:

Let I = 

⇒ I = 

[]

⇒ I = log |cosec π/4 – cot π/4| - log |cosec π/6 – cot π/6|

⇒ I = log |√2 – 1| - log |2 - √3|

∴  = 

Question 9.

Evaluate 

Answer:

Let I = 

⇒ I = []

⇒ I = 

⇒ I = sin^-1(1) – sin^-1(0) = π/2 – 0

⇒ I = π/2

∴  = π/2

Question 10.

Evaluate 

Answer:

Let I = 

⇒ I = 

We know that,

Therefore,

⇒ I = 

⇒ I = tan^-1(1) – tan^-1(0) = π/4 – 0

⇒ I = π/4

∴  = π/4

Question 11.

Evaluate 

Answer:

Let I = 

⇒ I = []

⇒ I =  = 

⇒ I =  = 

⇒ I = � log 3/2

∴  = � log 3/2

Question 12.

Evaluate 

Answer:

Let I = 

cos 2x = 2cos²x – 1

cos²x = 

putting the value cos²x in I

⇒ I = []

⇒ I = 

⇒ I =  = π/4

∴  = π/4

Question 13.

Evaluate 

Answer:

Let I = 

Let x² + 1 = t …(i)

∴ d(x² + 1) = dt

⇒ 2x dx = dt

⇒ x dx = dt/2

When x = 2; t = 2² + 1 = 5

When x = 3; t = 3² + 1 = 10

Substituting x² + 1 and x dx in I

⇒ I = []

⇒ I = 

⇒ I = � log 2

= � log 2

Question 14.

Evaluate 

Answer:

Let I = 

Multiplying by 5 in numerator and denominator

⇒ I = 

⇒ I = 

⇒ I = I₁ + I₂

I₁ = 

Let 5x² + 1 = t ….(i)

d(5x² + 1) = dt

10x dx = dt …..(ii)

When x = 0; t = 5 × 0² + 1 = 1

When x = 1; t = 5 × 1² + 1 = 6

Substituting (i) and (ii) in I₁

I₁ = []

I₁ = 

I₁ = 

I₂ = []

I₂ = 

I₂ = 3/√5 tan^-15

∵ I = I₁ + I₂

∴ I = 1/5 log 6 + 3/√5 tan^-15

∴  = 1/5 log 6 + 3/√5 tan^-15

Question 15.

Evaluate 

Answer:

Let I = 

Put x² = t ⇒ 2x dx = dt

When x = 0; t = 0

When x = 1; t = 1

Substituting t and dt in I

⇒ I =  = � []

⇒ I = 

∴  = � (e – 1)

Question 16.

Evaluate 

Answer:

Let I = 

Dividing 5x² by x² + 4x + 3 we get 5 as quotient and –(20x + 15) as remainder

So, I = 

⇒ I =  = 

⇒ I = 5 (2 – 1) - 

⇒ I = 5 – I₁

I₁ = 

Adding and subtracting 25 in the numerator

I₁ = 

I₁ = 

Let x² + 4x + 3 = t

(2x + 4)dx = dt

∴ I₁ = 

I₁ = 10 log t - []

I₁ = 10 log t - []

I₁ = 

I₁ = 10

I₁ = 

I₁ = 

I₁ = 

I₁ = 

I₁ = 

∵ I = 5 – I₁

Substituting I₁ in I we get

I = 5 – 

∴  = 5 – 

Question 17.

Evaluate 

Answer:

Let I = 

⇒ I = 

⇒ I = []

⇒ I = 2 (tan π/4 – tan 0) + � ((π/4)⁴ – 0) + 2 (π/4 – 0)

⇒ I = 

⇒ I = 

∴ 

Question 18.

Evaluate 

Answer:

Let I = 

We know, 

Substituting  in I, we get

⇒ I = []

⇒ I = 

∴  = 0

Question 19.

Evaluate 

Answer:

Let I = 

I = 

∴ I = I₁ + I₂

I₁ = 

Let x² + 4 = t

2x dx = dt

When x = 0; t = 4

When x = 2; t = 2² + 4 = 8

Substituting t and dt in I₁

⇒ I₁ = []

⇒ I₁ = 3 [log |8| - log |4|] = 3 log 8/4

⇒ I₁ = 3 log � = -3 log 2

I₂ =  = []

⇒ I₂ = 

⇒ I₂ =  = 3π/8

Now I = I₁ + I₂

I = 3 log � + 3π/8

∴  = 3 log � + 3π/8

Question 20.

Evaluate 

Answer:

Let I = 

⇒ I = 

I = I₁ + I₂

I₁ = 

⇒ I₁ = []

⇒ I₁ = 

⇒ I₁ = e – e – 0 + 1

⇒ I₁ = 1

I₂ = []

⇒ I₂ = 

⇒ I₂ =  = 

Since, I = I₁ + I₂

∴ I = 1 + 

∴  = 1 + 

Question 21.

 equals
A. π/3

B. 2π/3

C. π/6

D. π/12

Answer:

Let I = 

⇒ I = 

⇒ I = []

⇒ I = π/12

∴ π/12

Question 22.

 equals
A. π/6

B. π/12

C. π/24

D. π/4

Answer:

Let I = 

⇒ I = 

Taking 9 common from Denominator in I

⇒ I = []

⇒ I =

⇒ I = 

⇒ I =  = π/24

∴  = π/24