Question

Integral of sqrt of x ln x dx.. ln x is outside of sqrt.

Answer 1

$\displaystyle \int \sqrt x \ln x\, dx=(*)\\ t=\ln x,du=\sqrt x\, dx\\ dt=\dfrac{1}{x}\, dx, u=\dfrac{2}{3}x^{\tfrac{3}{2}}\\ (*)=\dfrac{2}{3}x^{\tfrac{3}{2}}\ln x-\int \dfrac{1}{x}\cdot\dfrac{2}{3}x^{\tfrac{3}{2}}\, dx=\\ \dfrac{2}{3}x^{\tfrac{3}{2}}\ln x-\dfrac{2}{3}\int \sqrt x\, dx=\\ \dfrac{2}{3}x^{\tfrac{3}{2}}\ln x-\dfrac{2}{3}\cdot\dfrac{2}{3}x^{\tfrac{3}{2}}=\\ \dfrac{2}{3}x^{\tfrac{3}{2}}\left(\ln x-\dfrac{2}{3}\right)+C$