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) 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

\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

in v erse l a pl a ce \frac{1}{s ( s - 1 ) ^{2}}

f (x) = - 3 x

im pl i c i t \frac{d y}{d x}, 25 x = y^{4}

a re a 6 x^{2}, x^{2} + 9

t an g e n t f (x) = 3 x^{2} - 2 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