Question

`int cos^3 (ax+b)sin(ax+b) dx` ? Please integrate this using substitution

Answer 1

`I=-1/(4a)cos^4(ax+b)+c`

Explanation:

Here,

`I=intcos^3(ax+b)*sin(ax+b)dx`

Subst. `cos(ax+b)=u=>-sin(ax+b)*adx=du`

`=>sin(ax+b)dx=-1/adu`

So,

`I=intu^3(-1/a)du`

`:.I=-1/aintu^3du`

`=-1/a*u^4/4+c`

`=-1/(4a)*u^4+c`

Subst. back ,`u=cos(ax+b)` , we get

`I=-1/(4a)cos^4(ax+b)+c`
Note:

Please write only "int" for "integration"-symbol.