Question

Integrate

$\int e^x\frac{\left(1+x^2\right)}{{\left(1+x\right)}^2}dx$

Answer 1

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

$ApplyIntegrationByParts:u=e^x(1+x^2),v^{\prime }=\frac{1}{{(1+x)}^2}$

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

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

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