Mathematics

Solve:
$$\dfrac{1}{2}\int {\dfrac{{{u^2}}}{{1 + u}}du} $$


SOLUTION
$$ I =\dfrac {1}{2} \int\dfrac {u^2 du}{1+u}$$

$$ =\dfrac {1}{2}\int  \dfrac {(u^2-1+1)du}{1+u}$$

$$=\dfrac {1}{2}\int  \dfrac {(u^2-1)}{1+u}du +\dfrac {1}{2}\int  \dfrac {1}{1+u}du$$

$$=\dfrac {1}{2}\int  \dfrac {(u+1)(u-1)}{(u+1)}du \,\, \dfrac {1}{2} \int  \dfrac {1}{u+1}du$$

$$ =\dfrac {1}{2}\left [ \dfrac{u^2}{2}-u \right ]+\dfrac {1}{2}log |u+1| +C$$

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