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

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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.

1 Answer
Dec 17, 2017

#-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)#.