What is the integral of sqrt(x)ln(9x) ?

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

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integral of sqrt(x)ln(9x)

\int x ln (9 x) d x

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

Solution steps

\int x ln (9 x) d x

Apply Integration By Parts

: \frac{2}{3} x^{\frac{3}{2}} ln (9 x) - \int \frac{2}{3} x^{\frac{1}{2}} d x

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

Add a constant to the solution

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

x \to 2 lim (x^{2} + 4)

\frac{d}{d x} (x e^{c o s (2 x - y)})

x \to + 3 + lim (\frac{x + 3}{x - 3})

\frac{\partial}{\partial y} (x^{3} + x^{2} y - y^{2} - 4 y)

\int_{- 1}^{2} (x - 1) (2 x + 3) d x

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