Doubt in integral substitution

Question

I am not able to figure out what substitution to use in the following integral $$ \int \frac{(x-1)e^x}{(x+1)^3}dx $$

Any help would be appreciated.

Answer 1

Hint: How about let $u = x + 1 $ first. And express the integral in $u$.

Answer 2

Hint

You can use integration by parts with $$u'=\frac{(x-1)}{(x+1)^3}=\frac{(x+1-2)}{(x+1)^3}=\frac{1}{(x+1)^2}-\frac{2}{(x+1)^3}$$ $$v=e^x$$ from which $$u=-\frac{x}{(x+1)^2}$$ $$v'=e^x$$ and a second integration by parts will lead you to the result.

I am sure that you can take from here.