What is the integral of (x^2+5)ln(x) ?

The integral of (x^2+5)ln(x) is ln(x)((x^3)/3+5x)-1/3 ((x^3)/3+15x)+C

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integral of (x^2+5)ln(x)

\int (x^{2} + 5) ln (x) d x

ln (x) (\frac{x ^{3}}{3} + 5 x) - \frac{1}{3} (\frac{x ^{3}}{3} + 15 x) + C

Solution steps

\int (x^{2} + 5) ln (x) d x

Apply Integration By Parts

: ln (x) (\frac{x ^{3}}{3} + 5 x) - \int \frac{1}{3} (x^{2} + 15) d x

= ln (x) (\frac{x ^{3}}{3} + 5 x) - \int \frac{1}{3} (x^{2} + 15) d x

\int \frac{1}{3} (x^{2} + 15) d x = \frac{1}{3} (\frac{x ^{3}}{3} + 15 x)

= ln (x) (\frac{x ^{3}}{3} + 5 x) - \frac{1}{3} (\frac{x ^{3}}{3} + 15 x)

Add a constant to the solution

= ln (x) (\frac{x ^{3}}{3} + 5 x) - \frac{1}{3} (\frac{x ^{3}}{3} + 15 x) + C

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What is the integral of (x^2+5)ln(x) ?
The integral of (x^2+5)ln(x) is ln(x)((x^3)/3+5x)-1/3 ((x^3)/3+15x)+C