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

The integral of sqrt(x^3)ln(2x) is 2/5 x^{5/2}ln(2x)-4/25 x^{5/2}+C

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

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

\frac{2}{5} x^{\frac{5}{2}} ln (2 x) - \frac{4}{25} x^{\frac{5}{2}} + C

Solution steps

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

x^{3} = x^{\frac{3}{2}},

assuming

x \geq 0

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

Apply Integration By Parts

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

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

\int \frac{2}{5} x^{\frac{3}{2}} d x = \frac{4}{25} x^{\frac{5}{2}}

= \frac{2}{5} x^{\frac{5}{2}} ln (2 x) - \frac{4}{25} x^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{5} x^{\frac{5}{2}} ln (2 x) - \frac{4}{25} x^{\frac{5}{2}} + C

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What is the integral of sqrt(x^3)ln(2x) ?
The integral of sqrt(x^3)ln(2x) is 2/5 x^{5/2}ln(2x)-4/25 x^{5/2}+C