Question

How do you evaluate the integral `int dx/(x+1)^3`?

Answer 1

`intdx/(x+1)^3=(-1)/(2(x+1)^2)+C`

Explanation:

Let `u=x+1`. Differentiating this shows that `du=dx`. We can then write:

`intdx/(x+1)^3=int(du)/u^3=intu^-3du`

This can be integrated through the rule `intu^ndu=u^(n+1)/(n+1)+C` where `n!=-1`.

`intdx/(x+1)^3=u^(-2)/(-2)+C=(-1)/(2u^2)+C`

Returning to the original variable `x` with `u=x+1`:

`intdx/(x+1)^3=(-1)/(2(x+1)^2)+C`