Question

Prove that
$\int \frac{e^x(2-x)}{(1-x{)}^2}dx$

Answer 1

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

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

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

Let t $=e^x\left(\frac{1}{1-x}\right)$

dt$=[e^x\left(\frac{1}{1-x}\right)+e^x\left(\frac{1}{(1-x{)}^2}\right)]dx$

$=\int dt=t+c$

$=\frac{e^x}{1-x}+c$