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

x \to 3 lim (3 x - 1)

\int cos^{2} (a) x d x

\int_{1}^{\infty} \frac{x}{x ^{2} + 3} d x

\int 1 - 16 x^{2} d x

arcsin (\frac{x}{2})

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