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=(1)/(x)\, dx, u=(2)/(3)x^{\tfrac{3}{2}}\\ (*)=(2)/(3)x^{\tfrac{3}{2}}\ln x-\int (1)/(x)\cdot(2)/(3)x^{\tfrac{3}{2}}\, dx=\\ (2)/(3)x^{\tfrac{3}{2}}\ln x-(2)/(3)\int \sqrt x\, dx=\\ (2)/(3)x^{\tfrac{3}{2}}\ln x-(2)/(3)\cdot(2)/(3)x^{\tfrac{3}{2}}=\\ (2)/(3)x^{\tfrac{3}{2}}\left(\ln x-(2)/(3)\right)+C`