Is this Expectation finite? How do I prove that
$\int_{0}^{+\infty}\text{exp}(-x)\cdot\text{log}(1+\frac{1}{x})dx$ 
is finite? (if it is)
I tried through simulation and it seems finite for large intervals. But I don't know how to prove it analytically because I don't know the closed form integral of this product.
I am actually taking expectation over exponential distribution.
Thank you

If we write $$\int_{0}^{+\infty}=\int_0^1+\int_1^{\infty}$$ then we see that $$\lim_{x\to 0^+} x^{1/2}f(x)=0<\infty\Longrightarrow\int_0^1f(x)dx~~\text{is convergent}$$ and $$\lim_{x\to +\infty} x^{2}f(x)=0<\infty\Longrightarrow\int_1^{\infty}f(x)dx~~\text{is convergent}$$
