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

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

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integral of ln(x)*sqrt(x)

\int ln (x) \cdot x d x

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

Solution steps

\int ln (x) 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

= \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}}

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

Add a constant to the solution

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

x \to 0 lim (\frac{e ^{x} - 1}{19 x})

x \to - 2 lim (3 x)

\frac{d y}{d x} = 7 x e^{y}, y (0) = 0

\frac{d}{d x} (- 3 cos^{2} (x) sin (x))

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

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