Let's integrate the following function:
Function: ∫ cos(x) dx
Steps:
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Recall the Integral of cos(x): The basic integral of cos(x) is sin(x). This is one of the standard trigonometric integrals that you should memorize.
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Add the Constant of Integration (C): Since indefinite integrals represent a family of functions, we add the constant of integration: ∫ cos(x) dx = sin(x) + C
Explanation:
- Integration is the reverse process of differentiation.
- The derivative of sin(x) is cos(x), so the integral of cos(x) is sin(x).
- The constant of integration 'C' accounts for any vertical shift of the antiderivative, since the derivative of a constant is zero.
Let's check our result:
Take the derivative of our answer, sin(x) + C:
- d/dx [sin(x) + C] = cos(x) + 0 = cos(x)
Since the derivative gets us back to the original function, our solution is correct!
Important Note: This is an indefinite integral (no limits of integration). Definite integrals, where we integrate over a specific interval, will result in a numerical answer instead of a function plus a constant.
Let me know if you'd like to try another example or work on a more complex integral!
