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Question Number 60797 by arcana last updated on 25/May/19

$$\int \frac{e^w}{w^{n+1}}dw,n\in \mathbb{N}$$

Commented by MJS last updated on 26/May/19

$$\mathrm{this}\mathrm{reminds}\mathrm{me}\mathrm{of}\mathrm{\Gamma }\left(x\right)=\underset{0}{\stackrel{\mathrm{\infty }}{\int }}{\mathrm{e}}^{-t}{t}^{x-1}dt$$$$\int {\mathrm{e}}^w{w}^{-n-1}dw=$$$$[v=w^{-n}\to dw=-\frac{w^{n+1}}{n}dv]$$$$=-\frac{1}{n}\int {\mathrm{e}}^{1/\sqrt[n]{v}}dv$$$$\mathrm{this}\mathrm{is}\mathrm{a}\mathrm{special}\mathrm{integral}\mathrm{called}incomplete$$$$gammafunction.\mathrm{I}\mathrm{found}\mathrm{this}\mathrm{online}$$$$\int {\mathrm{e}}^{1/\sqrt[n]{v}}dv={(-1)}^{n}n\mathrm{\Gamma }(-n,-\frac{1}{\sqrt[n]{v}})$$$$\Rightarrow -\frac{1}{n}\int {\mathrm{e}}^{1/\sqrt[n]{v}}dv=-{(-1)}^n\mathrm{\Gamma }(-n,-\frac{1}{\sqrt[n]{v}})=$$$$=-{(-1)}^n\mathrm{\Gamma }(-n,-w)+C$$$$\mathrm{sorry}\mathrm{cannot}\mathrm{further}\mathrm{explain}...$$

Commented by arcana last updated on 26/May/19

$$\mathrm{muchas}\mathrm{gracias},\mathrm{I}\mathrm{need}\mathrm{this}\mathrm{but}\mathrm{now}$$$$\mathrm{I}\mathrm{have}\mathrm{some}\mathrm{idea}:\mathrm{D}$$

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