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I'm a little rusty with my calculus so I was hoping if anyone would help me with the following integral:

$$\int_{-\infty}^{\infty} \prod _{i=1}^ne^{-(y-v_i)}e^{-e^{-(y-v_i)}}dy$$

Thanks in advance for your help.

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up vote 4 down vote accepted

Using $x=\mathrm e^{-y}$, the integral you are interested in is $$ I=\int_0^{+\infty}\prod_{i=1}^n\left(x\mathrm e^{v_i}\cdot\mathrm e^{-\mathrm e^{v_i}x}\right)\cdot\frac{\mathrm dx}x=\int_0^{+\infty}ax^{n-1}\mathrm e^{-bx}\mathrm dx, $$ for some suitable positive $a$ and $b$, depending on $(v_i)_{1\leqslant i\leqslant n}$, that I will let you discover. Thus, $$ I=\int_0^{+\infty}ab^{-n}x^{n-1}\mathrm e^{-x}\mathrm dx=(n-1)!\cdot ab^{-n}. $$

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