Question

Find: $\int \frac{(x^2+1)e^x}{{(1+x)}^2}dx$

Answer 1

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

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

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

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

$=e^x\left(\frac{x-1}{x+1}\right)+c$