What is the integral of sqrt(x)*ln^2(x) ?

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

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integral of sqrt(x)*ln^2(x)

\int x \cdot ln^{2} (x) d x

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

Solution steps

\int x ln^{2} (x) d x

Apply Integration By Parts

: \frac{2}{3} x^{\frac{3}{2}} ln^{2} (x) - \int \frac{4}{3} x^{\frac{1}{2}} ln (x) d x

= \frac{2}{3} x^{\frac{3}{2}} ln^{2} (x) - \int \frac{4}{3} x^{\frac{1}{2}} ln (x) d x

\int \frac{4}{3} x^{\frac{1}{2}} ln (x) d x = \frac{4}{3} (\frac{2}{3} x^{\frac{3}{2}} ln (x) - \frac{4}{9} x^{\frac{3}{2}})

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

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