What is the integral of x^2ln(2x^3) ?

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

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integral of x^2ln(2x^3)

\int x^{2} ln (2 x^{3}) d x

\frac{1}{3} ln (2) x^{3} + x^{3} ln (x) - \frac{x ^{3}}{3} + C

Solution steps

\int x^{2} ln (2 x^{3}) d x

Apply Integration By Parts

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

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

\int x^{2} d x = \frac{x ^{3}}{3}

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

Simplify

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

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

Add a constant to the solution

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

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