How do I integrate $$ \int^{\infty}_0 \sqrt x * \exp \left( \frac{-x}{2\theta} \right) dx$$ $\theta$ is a strictly positive constant.
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Let $x=2 \theta t^2$ to make $$I=\int\sqrt{x} e^{-\frac{x}{2 \theta }}\,dx=4 \sqrt{2}\, \theta ^{3/2} \int e^{-t^2} t^2\,dt$$
Now, integrate by parts $$\int e^{-t^2} t^2\,dt=-\frac{1}{2} e^{-t^2} t+\frac 12 \int e^{-t^2}\,dt=-\frac{1}{2} e^{-t^2} t+\frac{1}{4} \sqrt{\pi } \,\text{erf}(t) $$ Now, it is very simple to get the definite integral.
Claude Leibovici
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