Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIntegrals1 Mark
Question
Integrate the function $x (\log x)^2$
✓
Answer
Let $I=x(\log x)^{2}$
Integrating by parts, we get,
$I=\left[(\log x)^2 \int x d x-\int\left\{\left(\frac{d}{d x} (\log x)^2 \right) \int x d x\right\} d x\right]$
$= \left[\frac{x^{2}}{2}(\log x)^2-\int \frac{2 \log x }{x} \cdot \frac{x^{2}}{2} d x\right]$
$= \frac{x^{2}}{2}(\log x)^{2}- \int x \log x \cdot d x$
$= \frac{x^{2}}{2}(\log x)^{2} - [ \log x \int x dx -\int( \frac {d}{dx} (\log x)\int x dx) dx]$
$= \frac{x^{2}}{2}(\log x)^{2} - [ \frac {x²}{2}\log x - \int(\frac1x \cdot\frac{x^2}{2} ) dx$
$= \frac{x^{2}}{2}(\log x)^{2}-\frac{x^{2}}{2} \log x+\frac{x^{2}}{4}+C$
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