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

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

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integral of sqrt(x)ln(3x)

\int x ln (3 x) d x

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

Solution steps

\int x ln (3 x) d x

Apply Integration By Parts

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

= \frac{2}{3} x^{\frac{3}{2}} ln (3 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}}

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

Add a constant to the solution

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

\int \frac{5 x ^{2}}{( 144 + x ^{2} ) ^{2}} d x

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

x \to 0 + lim (\frac{4 x}{sin ( x )})

\frac{d y}{d x} - y = e^{x} y^{2}

(- 17, - 5), (2, - 18)

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