Intergrate the following.

Question

y = cos(x)(2sin(x)-1)^6

Answer 1

Use substitution, p=sin(x), dp=cos(x)dx

I=∫ydx
=∫cos(x)(2sin(x)-1)^6dx
=∫(2p-1)^6 dp

Substitute again, u=2p-1, du=2dp
I=∫u^6 (dp/2)
=u^7/7 ÷2 + C

Now back substitute u=2p-1=2sin(x)-1

I=(2sin(x)-1)^7 /14 +C