Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
Question
Integrate the function x (log x)2
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Answer
Let $I=x(\log x)^{2}$ Integrating by parts, we get, $I=\left[(\log x)^2 \int x d x-\int\left\{\left(\frac{d}{d x} (\log x)^2 \right) \int x d x\right\} d x\right]$ = $\left[\frac{x^{2}}{2}(\log x)^2-\int \frac{2 \log x }{x} \cdot \frac{x^{2}}{2} d x\right]$ = $\frac{x^{2}}{2}(\log x)^{2}- \int x \log x \cdot d x$ = $\frac{x^{2}}{2}(\log x)^{2} - [ log x ∫ x dx -∫( \frac {d}{dx} (log x)∫ x dx) dx]$ = $\frac{x^{2}}{2}(\log x)^{2} - [ \frac {x²}{2}\log x - ∫(\frac1x \cdot\frac{x^2}{2} ) dx$ = $\frac{x^{2}}{2}(\log x)^{2}-\frac{x^{2}}{2} \log x+\frac{x^{2}}{4}+C$
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