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

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

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integral of x^{3/2}ln(3x)

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

\frac{2}{5} x^{\frac{5}{2}} ln (3 x) - \frac{4}{25} x^{\frac{5}{2}} + C

Solution steps

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

Apply Integration By Parts

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

= \frac{2}{5} x^{\frac{5}{2}} ln (3 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 (3 x) - \frac{4}{25} x^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{5} x^{\frac{5}{2}} ln (3 x) - \frac{4}{25} x^{\frac{5}{2}} + C

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