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Integration by Parts

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Question

Evaluate $\int x^{2} log x d x$

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Solution

$\begin{matrix} W e h a v e I = \int x^{2} log x = log x . \frac{x^{3}}{3} - \int \frac{x^{3}}{3} . \frac{1}{x} d x [∴ \int u . v d x = v u - \int v d u] = \frac{x^{3}}{3} log x - \frac{1}{3} \int x^{2} d x = \frac{x^{3}}{3} log x - \frac{1}{3} \times \frac{x^{3}}{3} = \frac{x^{3}}{3} . log x - \frac{x^{3}}{9} + C I t i s t h e r e q u i r e d a n s w e r . \end{matrix}$

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