Apply integration by parts, ∫e^xx(1+x)^2dx = e^xx1+x ∫ e^xdx = e^xx1+x ( e^x) = e^xx1+x+e^x Computing the points ∫ _0^1e^xx(1+x)^2dx=e2 ...

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∫01ex.xx+12dx...

Question

$\int_{0}^{1} \frac{e^{x} . x}{(x + 1)^{2}} d x =$

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Solution

Apply integration by parts,
$\int \frac{e^{x} x}{{(1 + x)}^{2}} d x = - \frac{e^{x} x}{1 + x} - \int - e^{x} d x = - \frac{e^{x} x}{1 + x} - (- e^{x}) = - \frac{e^{x} x}{1 + x} + e^{x}$
Computing the points
$\int_{0}^{1} \frac{e^{x} x}{{(1 + x)}^{2}} d x = \frac{e}{2} - 1$

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