What is the integral of x(ln(x)-1) ?

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

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integral of x(ln(x)-1)

\int x (ln (x) - 1) d x

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

Solution steps

\int x (ln (x) - 1) d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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