Question

The value of the definite integral ${\int }_0^1{e}^{e^x}\left(1+x\cdot e^x\right)dx$ is equal to

• A. $e^e$
• B. $e^e-e$
• C. $e^e-1$
• D. $e$

Answer 1

Answer: (A)

Put $e^x=t\Rightarrow e^{x}dx=dt$
$x=\ln t$
$I=\int e^t\frac{(1+t\ln t)}{t}dt=\int e^t\left(\ln t+\frac{1}{t}\right)dt={e^t\ln t|}_1^e=e^e$