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

= \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

(\frac{1}{x ^{2}} - \frac{5}{x ^{4}}) (x + 7 x^{3})

x \to 6 + lim (\frac{x + 3}{6 - x})

\frac{1}{s ( s - 1 ) ^{2}}

\frac{d}{d x} (\frac{q ^{2} + 45 q + 875}{3}) = 0

f (x) = 8 x - 3 x^{2}, (- 3, - 51)

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