What is the integral of xln(1/x) ?

The integral of xln(1/x) is -1/2 x^2ln(x)+(x^2)/4+C

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integral of xln(1/x)

\int x ln (\frac{1}{x}) d x

- \frac{1}{2} x^{2} ln (x) + \frac{x ^{2}}{4} + C

Solution steps

\int x ln (\frac{1}{x}) d x

Apply Integration By Parts

: - \frac{1}{2} x^{2} ln (x) - \int - \frac{x}{2} d x

= - \frac{1}{2} x^{2} ln (x) - \int - \frac{x}{2} d x

\int - \frac{x}{2} d x = - \frac{x ^{2}}{4}

= - \frac{1}{2} x^{2} ln (x) - (- \frac{x ^{2}}{4})

Simplify

= - \frac{1}{2} x^{2} ln (x) + \frac{x ^{2}}{4}

Add a constant to the solution

= - \frac{1}{2} x^{2} ln (x) + \frac{x ^{2}}{4} + C

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What is the integral of xln(1/x) ?
The integral of xln(1/x) is -1/2 x^2ln(x)+(x^2)/4+C