Question

$\int$x^x(1 + log x) dx = ________________.

Answer 1

$I=\int x^x\left(1+\log x\right)dx$

Let $x^x=z$

$\Rightarrow \log z=\log x^x$

$\Rightarrow \log z=x\log x$

$\Rightarrow \frac{1}{z}dz=\left(x\times \frac{1}{x}+\log x\times 1\right)dx$

$\Rightarrow \frac{dz}{z}=\left(1+\log x\right)dx$

$\therefore I=\int z\times \frac{dz}{z}$

$\Rightarrow I=\int dz$

$\Rightarrow I=z+C$

$\Rightarrow I=x^x+C$


$\int$x^x(1 + log x)dx = ____x^x + C____.