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

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

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integral of 3sqrt(x)ln(x)

\int 3 x ln (x) d x

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

Solution steps

\int 3 x ln (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int x ln (x) d x

Apply Integration By Parts

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

\int \frac{2}{3} x^{\frac{1}{2}} d x = \frac{4}{9} x^{\frac{3}{2}}

= 3 (\frac{2}{3} x^{\frac{3}{2}} ln (x) - \frac{4}{9} x^{\frac{3}{2}})

Simplify

3 (\frac{2}{3} x^{\frac{3}{2}} ln (x) - \frac{4}{9} x^{\frac{3}{2}}) : 2 x^{\frac{3}{2}} ln (x) - \frac{4}{3} x^{\frac{3}{2}}

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

Add a constant to the solution

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

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