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

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

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integral of-ln(x)x

\int - ln (x) x d x

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

Solution steps

\int - ln (x) x d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int ln (x) x d x

Apply Integration By Parts

= - (\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

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