Question

Object: $\int \left(ex{)}^x\right(2+logx)dx$.

Answer 1

$I=\int \left(ex{)}^x\right(2+logx)dx$
$=\int e^x\cdot x^x(2+logx)dx$
$=\int e^x{x}^x[1+(1+logx\left)\right]dx$
$=\int e^x[x^x+x^x(1+logx\left)\right]dx$
Now taking $f\left(x\right)=x^x$
$\therefore f^{\prime }\left(x\right)=x^x(1+logx)+c$
$\therefore I=\int e^x\left(f\right(x)+f^{\prime }(x\left)\right)dx$
$=e^{x}f\left(x\right)+v$
$=e^x\cdot x^2+c$
$I=(e^x{)}^x+c$.