This is an extract from text book, I need help understanding how to integrate the function at line 3. Point me in the right direction please.
2026-04-02 18:14:31.1775153671
Distribution with random parameter
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Notice that the expression is just the pdf of gamma distribution with parameter $\beta=2$ and $\alpha=k+1$.
$$f(x; \alpha, \beta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}.$$
Hence it integrates to $1$.