Question

Solve : $\underset{0}{\stackrel{1}{\int }}{e}^{e^x}(1+x.e^x)dx$

Answer 1

${\int }_0^1{e}^{e^x}(1+x{e}^x)dx$

substitute $u=e^x$

$\int \frac{e^u(1+\ln (u\left)u\right)}{u}du$

$\int \frac{e^u}{u}+e^u\ln \left(u\right)$

$Ei\left(u\right)+e^u\ln \left(u\right)-Ei\left(u\right)$

$e^u\ln \left(u\right)$

$e^{e^x}\ln \left(e^x\right)$

$e^{e^x}x+C$

${\int }_0^1{e}^{e^x}(1+x{e}^x)dx$

$e^e-0$

$e^e$