Question

Find :
$\int \frac{e^x(x-1)}{(x+1{)}^3}dx$

Answer 1

Find

$\int e^x\frac{(x-1)}{{(x-1)}^3}dx$

$\int e^x\frac{x+1-2}{{(x-1)}^3}dx$

$\int e^x\{\frac{x+1}{{(x+1)}^3}-\frac{2}{{(x+1)}^3}\}dx$

$=\int \{\frac{e^x}{{(x+1)}^2}-\frac{2e^x}{{(x+1)}^3}\}dx$

$\Rightarrow \frac{e^x{(x+1)}^2-e^{x}2(x+1)dx}{{(x+1)}^4}=dz$

Or $\{\frac{e^x}{{(x+1)}^2}-\frac{2e^x}{{(x+1)}^3}\}dx=dz$

$\int dz=z+c$

$=\frac{e^x}{{(x+1)}^2}+c$

Ans $=\frac{e^x}{{(x+1)}^2}$