Question

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

Answer 1

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

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

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

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

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

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

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

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