What is the integral using the substitution u=sin(2x)?

Question

find $int_0^(pi/4)sin^4(2x)cos(2x)dx$ using the substitution $u=sin(2x)$

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Answer 1 · 2018-05-29T06:31:48

$I=1/10$

We want to integrate

$I=int_0^(pi/4)sin^4(2x)cos(2x)dx$

Make a substitution $color(blue)(u=sin(2x)=>du=2cos(2x)dx$
New limits $color(blue)(x=0=>u=0$ and $color(blue)(x=pi/4=>u=1$

$I=1/2int_0^1u^4du=[1/10u^5]_0^1=1/10$