Question

How do you find the definite integral of cos^4(2x)sin(2x) dx?

Please help guide me through this question with steps. I just want to know how to approach this question. I tried using u-substitution with u = 2x but I am unable to find the answer. Thanks for your time.

Answer 1

`-1/5`

Explanation:

Remember that the point of u-substitutions is to eliminate complex polynomials or functions. Under this mentality it would make sense that we try to eliminate the trigonometric functions sine and cosine. We will let `u = cos(2x)`, so that `du = -2sin(2x) dx`.

`int_1^(-1) " "u^4 sin(2x) * (du)/(-2sin(2x))`

Notice how the `sin(2x)`s cancel out, making our u-subsitution very useful. We are now left with

`int_1^(-1)` `-1/2u^4` `du`

which yields `-1/10u^5` evaluated from 1 to -1. Thus, our final answer is `color(red)(-1/5)`.