Evaluate intcos^5x dx using u-substitution or tabular integration?

1 Answer
Jarob R G.
Apr 30, 2018

sinx - 2/3 sin^3x + 1/5 sin^5x + C

Explanation:

We will use u-substitution and a trigonometric identity.

Recall that cos^2 x + sin^2 x = 1. Rearranging, it follows that cos^2 x = 1 - sin^2 x. Note that we can use this fact to change the integral as follows:

int cos^5 x dx = int (cos^2x)^2 cosx dx
= int (1-sin^2 x)^2 cosxdx

Make a u-substitution by letting u = sinx. Then du = cosx dx. Our integral becomes:

int (1-u^2)^2 du = int (1 - 2u^2 + u^4) du
= u - 2/3 u^3 + 1/5 u^5 + C
= sinx - 2/3 sin^3x + 1/5sin^5x + C

This is our final answer.

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