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

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

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integral of sqrt(x)*ln(x)

\int x \cdot ln (x) d x

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

Solution steps

\int x ln (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

\int_{- 4}^{0} (1 + 16 - x^{2}) d x

\int t^{2} 13 t - 1 d t

f (x) = (\frac{4}{x}) + (\frac{1}{5 x ^{3} + 2 x})

\int_{0}^{\frac{π}{2}} sec^{2} (\frac{x}{2}) d x

x y^{^{'}} - 2 y = 0

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