What is the integral of x(1+2ln(x)) ?

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

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integral of x(1+2ln(x))

\int x (1 + 2 ln (x)) d x

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

Solution steps

\int x (1 + 2 ln (x)) d x

Apply Integration By Parts

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

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

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

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

Expand

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

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

Add a constant to the solution

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

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