# Find integral of (((x^2)+1).e^x)/(x+1)^2  With respect to x?

Apr 7, 2018

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

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

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

$\implies \int {e}^{x} \left[\frac{x - 1}{x + 1}\right] \mathrm{dx} + 2 \int {e}^{x} \left[\frac{1}{x + 1} ^ 2\right] \mathrm{dx}$

Notice,
Considering $f \left(x\right) = \left[\frac{x - 1}{x + 1}\right]$

$f ' \left(x\right) = \frac{\left(x + 1\right) \left(x - 1\right) ' - \left(x - 1\right) \left(x + 1\right) '}{x + 1} ^ 2$

$\implies \frac{\left(x + 1\right) - \left(x - 1\right)}{x + 1} ^ 2 \implies \frac{2}{x + 1} ^ 2$

So the integral is now of the form,

$\implies \int {e}^{x} \left[f \left(x\right)\right] \mathrm{dx} + \int {e}^{x} \left[f ' \left(x\right)\right] \mathrm{dx}$

$\implies f \left(x\right) + c$

$\implies {e}^{x} \left[\frac{x - 1}{x + 1}\right] + c$