Consider the given integral. I=∫ x.log(x+1)dx I=log(x+1) .x^22 ∫1(x+1)x^22dx I=log(x+1) .x^22 ∫1(x+1)(x^2 1+1)2dx I=log(x+1) .x^22 12∫[ (x 1) ...

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Integration by Parts

∫ x.logx+1dx

Question

$\int x . l o g (x + 1) d x$

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Solution

Consider the given integral.
$I = \int x . l o g (x + 1) d x$
$I = l o g (x + 1) . \frac{x^{2}}{2} - \int \frac{1}{(x + 1)} \frac{x^{2}}{2} d x$
$I = l o g (x + 1) . \frac{x^{2}}{2} - \int \frac{1}{(x + 1)} \frac{(x^{2} - 1 + 1)}{2} d x$
$I = l o g (x + 1) . \frac{x^{2}}{2} - \frac{1}{2} \int [(x - 1) + \frac{1}{x + 1}] d x$
$I = l o g (x + 1) . \frac{x^{2}}{2} - \frac{1}{2} [\frac{x^{2}}{2} - x + l o g (x + 1)] + C$
where $c$ is the constant of integration.

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