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

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

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

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

\frac{1}{3} x^{3} ln (2 x) - \frac{x ^{3}}{9} + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{x ^{2}}{3} d x = \frac{x ^{3}}{9}

= \frac{1}{3} x^{3} ln (2 x) - \frac{x ^{3}}{9}

Add a constant to the solution

= \frac{1}{3} x^{3} ln (2 x) - \frac{x ^{3}}{9} + C

e^{x^{2}} - 1

\int (\frac{x}{2})^{4} d x

y^{^{'}} = \frac{y}{x}

\frac{\partial}{\partial y} (\frac{1}{ln ( 10 ) ( x - y )})

\frac{d y}{d t} + 3 y + e^{t} + 3 = 0

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