Question

Let $x$ be a continous variable defined over the interval $(-\infty ,\infty )$ and $f\left(x\right)=e^{-x-e^{-x}}$. The integral $g\left(x\right)=\int f\left(x\right)dx$ is equal to

• A. $e^{e^{-x}}$
• B. $e^{-e^{-x}}$
• C. $e^{-e^x}$
• D. $e^{-x}$

Answer 1

The correct option is B $e^{-e^{-x}}$

$g\left(x\right)=\int f\left(x\right)dx$
$f\left(x\right)=e^{-x-e^{-x}}$
$g\left(x\right)=\int e^{-x-e^{-x}}dx=\int \frac{e^{-x}}{e^{e^{-x}}}dx$
Substitute $e^{-x}=t$
$-e^{-x}dx=dt$
$g\left(x\right)=\int -\frac{dt}{e^t}=\int -e^{-t}dt$
$g\left(x\right)=e^{-t}$
$g\left(x\right)=e^{-e^{-x}}$