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

The integral of ((x^3)ln(2x)) is 1/4 x^4ln(2x)-(x^4)/(16)+C

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

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

\frac{1}{4} x^{4} ln (2 x) - \frac{x ^{4}}{16} + C

Solution steps

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

Apply Integration By Parts

: \frac{1}{4} x^{4} ln (2 x) - \int \frac{x ^{3}}{4} d x

= \frac{1}{4} x^{4} ln (2 x) - \int \frac{x ^{3}}{4} d x

\int \frac{x ^{3}}{4} d x = \frac{x ^{4}}{16}

= \frac{1}{4} x^{4} ln (2 x) - \frac{x ^{4}}{16}

Add a constant to the solution

= \frac{1}{4} x^{4} ln (2 x) - \frac{x ^{4}}{16} + C

10 \frac{d y}{d x} + 20 y = 12

\frac{d}{d x} (\frac{x ^{3}}{9})

\int_{2}^{6} \frac{x}{x - 2} d x

x^{4} + 8 x^{2} - x

f^{^{'}} (x) = x^{3} + 6 x^{2}

What is the integral of ((x^3)ln(2x)) ?
The integral of ((x^3)ln(2x)) is 1/4 x^4ln(2x)-(x^4)/(16)+C