Integrate the function $x^2\sqrt{1-x^2}$ from 0 to $1$

Find the integral

$\int\sin\ln\left(x\right)dx$

Find the integral

$\int\left(x^3\right)\log x^2dx$

Find the integral

$\int x^3\log\left(x\right)dx$

Find the integral

$\int\ln\left(\sqrt{x}+\sqrt{1+x}\right)dx$

Find the integral

$\int x^2\ln Xdx$

Find the integral

$\int x^2\:lnx\:dx$

Find the integral

$\int\:x\:ln\:2x\:dx$