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

The integral of (2x^2+1)ln(x) is 2/3 x^3ln(x)+xln(x)-(2x^3)/9-x+C

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

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

\frac{2}{3} x^{3} ln (x) + x ln (x) - \frac{2 x ^{3}}{9} - x + C

Solution steps

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

Apply Integration By Parts

: ln (x) (\frac{2 x ^{3}}{3} + x) - \int \frac{1}{3} (2 x^{2} + 3) d x

= ln (x) (\frac{2 x ^{3}}{3} + x) - \int \frac{1}{3} (2 x^{2} + 3) d x

\int \frac{1}{3} (2 x^{2} + 3) d x = \frac{2 x ^{3}}{9} + x

= ln (x) (\frac{2 x ^{3}}{3} + x) - (\frac{2 x ^{3}}{9} + x)

Simplify

ln (x) (\frac{2 x ^{3}}{3} + x) - (\frac{2 x ^{3}}{9} + x) : \frac{2}{3} x^{3} ln (x) + x ln (x) - \frac{2 x ^{3}}{9} - x

= \frac{2}{3} x^{3} ln (x) + x ln (x) - \frac{2 x ^{3}}{9} - x

Add a constant to the solution

= \frac{2}{3} x^{3} ln (x) + x ln (x) - \frac{2 x ^{3}}{9} - x + C

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What is the integral of (2x^2+1)ln(x) ?
The integral of (2x^2+1)ln(x) is 2/3 x^3ln(x)+xln(x)-(2x^3)/9-x+C