Question

How do you use substitution to integrate `(2x+3)/(x+7)^3`?

Answer 1

`int (2x+3)/((x+7)^3)\ dx=(-17-4x)/(2(x+7)^2)+C`

Explanation:

Let `u=x+7` so that `du=dx` and `x=u-7`. Then `\int (2x+3)/((x+7)^3)\ dx=int (2(u-7)+3)/u^3\ du=int (2u^{-2}-11u^{-3})\ du`

`=-2u^{-1}+(11/2)u^{-2}+C`

`=(-2u+11/2)/u^2+C`

`=(-4(x+7)+11)/(2(x+7)^2)+C=(-17-4x)/(2(x+7)^2)+C`