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

The integral of sqrt(3x)ln(x) is sqrt(3)(2/3 x^{3/2}ln(x)-4/9 x^{3/2})+C

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integral of sqrt(3x)ln(x)

\int 3 x ln (x) d x

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

Solution steps

\int 3 x ln (x) d x

3 x = 3 x

= \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

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

= 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}})

Add a constant to the solution

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

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What is the integral of sqrt(3x)ln(x) ?
The integral of sqrt(3x)ln(x) is sqrt(3)(2/3 x^{3/2}ln(x)-4/9 x^{3/2})+C