Question
Integrate the function x log x
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Answer
Let $ I = x log x$
Now, integrating by parts, we get,
Taking, Logarithmic function as first function and algebraic function as second function,
$I=\log x \int x d x-\int\left\{\left(\frac{d}{d x} \log x\right) \int x d x\right\} d x$
= $\log x\left(\frac{x^{2}}{2}\right)-\int \frac{1}{x} \cdot \frac{x^{2}}{2} d x$
= $\frac{x^{2} \log x}{2}-\int \frac{x}{2} d x$
= $\frac{x^{2} \log x}{2}-\frac{x^{2}}{4}+C$
Now, integrating by parts, we get,
Taking, Logarithmic function as first function and algebraic function as second function,
$I=\log x \int x d x-\int\left\{\left(\frac{d}{d x} \log x\right) \int x d x\right\} d x$
= $\log x\left(\frac{x^{2}}{2}\right)-\int \frac{1}{x} \cdot \frac{x^{2}}{2} d x$
= $\frac{x^{2} \log x}{2}-\int \frac{x}{2} d x$
= $\frac{x^{2} \log x}{2}-\frac{x^{2}}{4}+C$
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