So I have a double power law equation as a way to model the surface density $\Sigma$ of a debris disk as a function of radius $R$,
$$\Sigma(R)=\Sigma_c[(\frac{R}{r_c})^{-\alpha_\text{in} \gamma} + (\frac{R}{r_c})^{-\alpha_\text{out} \gamma}]^{-\frac{1}{\gamma}}$$
where $\alpha_\text{in}$ and $\alpha_\text{out}$ are the slopes in the inner and outer portion of the disk, $r_c$ the transition radius between them, $\gamma$ the smoothness of the transition, and $\Sigma_c$ the surface density at $r_c$.
The goal is to find an expression that, when given a certain mass $M$, tells me $\Sigma_c$, given a set of parametres $\alpha_\text{in}$, $\alpha_\text{out}$, $r_c$, and $\gamma$.
$$M=\int_0^\infty 2\pi R\Sigma(R)dR$$ $$=\int_0^\infty 2\pi R(\Sigma_c[(\frac{R}{r_c})^{-\alpha_\text{in} \gamma} + (\frac{R}{r_c})^{-\alpha_\text{out} \gamma}]^{-\frac{1}{\gamma}})dR$$
This is where my brain short-circuited. I couldn't get Mathematica to cooperate, and my calculus knowledge is near-obsolete, so if someone could walk me through how to go about this, I'd really appreciate it. Please let me know if there's any further context necessary that I've neglected to provide.