Integration using maple I am trying to evaluate the following integral using maple but it returns the unsolved integral. Can anybody help me in using maple to solve this integral?
$$
\int {2\pi\lambda R e^{-\lambda \pi R^2}} D^{a\pi R^2} (a \pi R^2)^C dR
$$
 A: Regrouping and ignoring non-essential constants, the integral can be rewritten
$$\int R^{2c+1}\alpha^{R^2}dR.$$
And by the change of variable $t=R^2$:
$$\int t^c\alpha^tdt.$$
Maple should recognize the exponential integral function.
A: The antiderivative is $$\pi ^{C+1} \lambda  \left(-a^C\right) R^{2 C+2} E_{-C}\left(\pi  R^2
   (\lambda -a \log (D))\right)$$
where $E_n(x)$ is a (generalized) exponential integral.  It is important that the argument to $E_{-C}$ be positive (otherwise certain steps in the calculation are potentially invalid).  With no bounds on the parameters in your integral, I can't determine whether this happened.  See here for an exponential integral implementation in Maple, where it would be $Ei(-C,\dots)$.
A: Maple finds it, when putting D (which is reserved for the differentiation operator) to be a local variable and the other parameters to be real:
$$local D; int(2*Pi*R*exp(-lambda*Pi*R^2)*D^{a*Pi*R^2}*(a*Pi*R^2)^C, R) $$  $$\, assuming \, real$$ outputs that,
 where the $\Gamma$ function is described here.
