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

The integral of sqrt(x)ln(sqrt(x)) is 1/3 x^{3/2}ln(x)-2/9 x^{3/2}+C

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integral of sqrt(x)ln(sqrt(x))

\int x ln (x) d x

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

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

\int \frac{1}{3} x^{\frac{1}{2}} d x = \frac{2}{9} x^{\frac{3}{2}}

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

Simplify

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

= \frac{1}{3} x^{\frac{3}{2}} ln (x) - \frac{2}{9} x^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{3} x^{\frac{3}{2}} ln (x) - \frac{2}{9} x^{\frac{3}{2}} + C

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What is the integral of sqrt(x)ln(sqrt(x)) ?
The integral of sqrt(x)ln(sqrt(x)) is 1/3 x^{3/2}ln(x)-2/9 x^{3/2}+C