Derive a reductions formula for integral emx/xn dx , where m and n are constants.
I=∫emxxndxPut z=x1−n ⟹ dz=1−nxndxI=−11−n∫emz11−ndzPut w=m1−nz ⟹ dw=m1−n=mn−11−n∫ew11−ndwBy using gamma function,=−mn−11−n(1−n)(−1)n−1Γ(1−n,−w11−n)=mn−1(−1)nΓ(1−n,−mx)+CI=\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)+CI=∫xnemxdxPut z=x1−n⟹dz=xn1−ndxI=−1−n1∫emz1−n1dzPut w=m1−nz⟹dw=m1−n=1−nmn−1∫ew1−n1dwBy using gamma function,=−1−nmn−1(1−n)(−1)n−1Γ(1−n,−w1−n1)=mn−1(−1)nΓ(1−n,−mx)+C
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