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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

\int x + 1 d x = \frac{x ^{2}}{2} + x

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

Simplify

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

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

Add a constant to the solution

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

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