Answer to Question #200592 in Calculus for Luc

Question

Derive a reductions formula for integral e^mx/xⁿ dx , where m and n are constants.

Answer 1

$I=\int \frac{e^{mx}}{x^n}dx\newline Put \space z=x^{1-n} \implies dz=\frac{1-n}{x^n}dx\newline I=-\frac{1}{1-n}\int e^{mz^{\frac{1}{1-n}}} dz \newline Put \space w=m^{1-n}z \implies dw=m^{1-n} \newline =\frac{m^{n-1}}{1-n}\int e^{w\frac{1}{1-n}} dw \newline \text{By using gamma function,} \newline =-\frac{m^{n-1}}{1-n} (1-n)(-1)^{n-1}\Gamma(1-n, -w^\frac{1}{1-n}) \newline =m^{n-1}(-1)^{n}\Gamma(1-n, -mx)+C$