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

The integral of sqrt(x)*ln(x^2) is 4/3 x^{3/2}ln(x)-8/9 x^{3/2}+C

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

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

\frac{4}{3} x^{\frac{3}{2}} ln (x) - \frac{8}{9} x^{\frac{3}{2}} + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{4}{3} x^{\frac{1}{2}} d x = \frac{8}{9} x^{\frac{3}{2}}

= \frac{2}{3} x^{\frac{3}{2}} ln (x^{2}) - \frac{8}{9} x^{\frac{3}{2}}

\frac{2}{3} x^{\frac{3}{2}} ln (x^{2}) = \frac{4}{3} x^{\frac{3}{2}} ln (x)

= \frac{4}{3} x^{\frac{3}{2}} ln (x) - \frac{8}{9} x^{\frac{3}{2}}

Add a constant to the solution

= \frac{4}{3} x^{\frac{3}{2}} ln (x) - \frac{8}{9} x^{\frac{3}{2}} + C

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What is the integral of sqrt(x)*ln(x^2) ?
The integral of sqrt(x)*ln(x^2) is 4/3 x^{3/2}ln(x)-8/9 x^{3/2}+C