What is the integral of sqrt(x)ln(2x) ?

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

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integral of sqrt(x)ln(2x)

\int x ln (2 x) d x

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

Solution steps

\int x ln (2 x) d x

Apply Integration By Parts

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

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

Add a constant to the solution

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

(\frac{2}{s} - \frac{1}{s ^{3}})^{2}

y = x + 3, y = \frac{( x + 3 )}{2}

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

\frac{\partial}{\partial x} (8 y)

\int x (x^{2} + 4) d x

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