# Convolution of probability densities. Having trouble integrating over vector

First, define a probability distribution with density:


I'm having trouble finishing this integral (to calculate the convolution):


Edit:

After looking at mpiktas's answer, the final line should have been \begin{align*} &= \prod_{i=1}^k\frac{\exp(b_il_i) - \exp(a_il_i)}{a_i-b_i} \end{align*} which ensures that the terms are all positive.

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However a lot of questions still remain. What is $B(\alpha+1)$? And if your are trying to compute convolution, why isn't your integration domain $(-\infty,0]^k$?