What is the integral of 2ln(x)x ?

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

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integral of 2ln(x)x

\int 2 ln (x) x d x

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

Solution steps

\int 2 ln (x) x d x

Take the constant out:

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

= 2 \cdot \int ln (x) x d x

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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x^{^{'}} = x^{2}

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\frac{1}{4 + x}