Question

What is the integral of sin^4(2x+3)cos^3(2x+3)?

Answer 1

`I = 1/10sin^5(2x+ 3) - 1/14sin^7(2x+ 3) + C`

Explanation:

Let `u = 2x+ 3`. Then `du = 2dx` and `dx = (du)/2`.

`I = 1/2int sin^4u cos^3u du`

`I = 1/2intsin^4ucos^2ucosu du`

`I = 1/2intsin^4u(1 - sin^2u)cosudu`

`I = 1/2intsin^4ucosu - sin^6u cosu du`

`I = 1/2int sin^4u cosu du - 1/2int sin^6u cosu du`

Let `t = sinu` then `dt = cosu du` and `du = (dt)/(cosu)`

`I = 1/2 int t^4 dt - 1/2 int t^6 dt`

`I = 1/2(1/5t^5) - 1/2(1/7t^7) + C`

`I = 1/10sin^5u - 1/14sin^7u + C`

`I = 1/10sin^5(2x+ 3) - 1/14sin^7(2x+ 3) + C`

Hopefully this helps!