I =12∫u^2 du1+u =12∫ #160; (u^2 1+1)du1+u =12∫ #160; (u^2 1)1+udu +12∫ #160; 11+udu =12∫ #160; (u+1)(u 1)(u+1)du 12∫ #160; 1u+1d ...

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Power of a Power

Solve: 12∫u2...

Question

Solve:
$\frac{1}{2} \int \frac{u^{2}}{1 + u} d u$

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Solution

$I = \frac{1}{2} \int \frac{u^{2} d u}{1 + u}$

$= \frac{1}{2} \int \frac{(u^{2} - 1 + 1) d u}{1 + u}$

$= \frac{1}{2} \int \frac{(u^{2} - 1)}{1 + u} d u + \frac{1}{2} \int \frac{1}{1 + u} d u$

$= \frac{1}{2} \int \frac{(u + 1) (u - 1)}{(u + 1)} d u \frac{1}{2} \int \frac{1}{u + 1} d u$

$= \frac{1}{2} [\frac{u^{2}}{2} - u] + \frac{1}{2} l o g | u + 1 | + C$

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