Solve \int x \ln x d x

Evaluate

\int x ln (x) d x

Prepare for integration by parts

u = ln (x) d v = x d x

Calculate the derivative

More Steps

d u = \frac{1}{x} d x d v = x d x

Evaluate the integral

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d u = \frac{1}{x} d x v = \frac{x ^{2}}{2}

Substitute u = ln (x) 、 v = \frac{x ^{2}}{2} 、 d u = \frac{1}{x} d x 、 d v = x d x for \int u d v = uv - \int v d u

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

Calculate

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

Evaluate the integral

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

Simplify the expression

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

Solution

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