The value of $$\int_0^\infty t^{-3/2}\left(1-e^{-t}\right)\,dt=$$
Applying integration by parts or using any kind of substitution is not working.
My attempt:
Splitting the integrands and integrating t^-1.5 over 0 to infinity gives infinity.
Now, treating t^-1.5 and (1- exp(-t)) as two separate functions and applying integration by parts also results in same problem.
Ps: This is not a homework question.
Integration by parts gives $$ -\left.2t^{-1/2}(1-e^{-t})\right|_0^{\infty}+2\int_0^{\infty}t^{-1/2}e^{-t}\ dt. $$ The first term vanishes both at $0$ and $\infty$ (observe that $1-e^{-t}$ vanishes to first order at $0$). To evaluate the second term, consider the substitution $u=\sqrt t$ to express it in terms of the integral $$ \int_0^{\infty}e^{-u^2}\ du. $$ This is a famous integral, and the trick to evaluating it is to square it then convert to polar coordinates. This is where the $\sqrt{\pi}$ comes from.