Question

What is the integral of `int sin^3(x) cos^5(x) dx`?

Answer 1

`int sin^3 x* cos^5 x* d x=-1/6 cos^6 x+1/8 cos^8 x+C`

Explanation:

`int sin^3 x* cos^5 x* d x=int sin x*sin^2 x* cos^5 x* d x`
`sin^2 x=1-cos^2 x`
`int color(red)(sin x)(1-cos^2 x) cos^5 x *color(red)(d x)`
`u=cos x" "d u=-sin x*d x`
`-int (cos^5 x-cos^7 x)d u=-int(u⁵-u^7)d u`
`int sin^3 x* cos^5 x* d x=-int u^5 d u +int u^7 d u+C`
`int sin^3 x* cos^5 x* d x=-1/6u^6+1/8u^8+C`
`int sin^3 x* cos^5 x* d x=-1/6 cos^6 x+1/8 cos^8 x+C`