Question

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

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

Answer 1

`I=1/10`

Explanation:

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`