First slide

Introduction to integration

Question

$Evaluate \int x \log x d x$

Easy

A

$\frac{x^{2}}{4} \log x - \frac{1}{4} x^{2} + C$
B

$\frac{x^{2}}{2} \log x - \frac{1}{2} x^{2} + C$
C

$\frac{x^{2}}{2} \log x - \frac{1}{4} x^{2} + C$
D

None of these

Solution

$\begin{array}{l} \int_{II} x \log x d x \\ = \log x \{\int x d x\} - \int \{\frac{d}{d x} (\log x) \int x d x\} d x \\ = (\log x) \frac{x^{2}}{2} - \int \frac{1}{x} \frac{x^{2}}{2} d x \\ = \frac{x^{2}}{2} \log x - \frac{1}{2} \int x d x \\ = \frac{x^{2}}{2} \log x - \frac{1}{2} (\frac{x^{2}}{2}) + C = \frac{x^{2}}{2} \log x - \frac{1}{4} x^{2} + C \end{array}$

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