Integrating the Schechter function...

Question

I'm trying to integrate this equation over all L.

I really have no idea where to start for some reason :S

$$\phi(L)dL=\phi_0\left(\dfrac L{L\star}\right)^\alpha\exp\left(-\dfrac L{L\star}\right)\dfrac{dL}{L\star}.$$

According to the internet I should end up with this:

$$\int_0^\infty L\phi d=\phi_0L_\star\Gamma(\alpha+2)$$

But I can't understand why

Answer 1

As suggested in the comment the integral is easy to solve in terms of the Gamma function:

Rescale $L/L\star=x$, $(L\star)dx=L$ \begin{align} \int_0^{\infty}\phi_0\left(\frac{L}{L\star}\right)^{\alpha}e^{-\frac{L}{L\star}}\frac{dL}{L\star}=(\phi_0L\star)\int_0^{\infty}t^{\alpha+1}e^{-x}dx=\phi_0(L\star)\Gamma(\alpha+2) \end{align}