What is the integral of x^{-3}e^x ?

The integral of x^{-3}e^x is -(e^x}{2x^2}+1/2 (-\frac{e^x)/x+\Ei(x))+C

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integral of x^{-3}e^x

\int x^{- 3} e^{x} d x

- \frac{e ^{x}}{2 x ^{2}} + \frac{1}{2} (- \frac{e ^{x}}{x} + Ei (x)) + C

Solution steps

\int x^{- 3} e^{x} d x

Simplify

x^{- 3} e^{x} : \frac{e ^{x}}{x ^{3}}

= \int \frac{e ^{x}}{x ^{3}} d x

Apply Integration By Parts

: - \frac{e ^{x}}{2 x ^{2}} - \int - \frac{e ^{x}}{2 x ^{2}} d x

= - \frac{e ^{x}}{2 x ^{2}} - \int - \frac{e ^{x}}{2 x ^{2}} d x

\int - \frac{e ^{x}}{2 x ^{2}} d x = - \frac{1}{2} (- \frac{e ^{x}}{x} + Ei (x))

= - \frac{e ^{x}}{2 x ^{2}} - (- \frac{1}{2} (- \frac{e ^{x}}{x} + Ei (x)))

Simplify

= - \frac{e ^{x}}{2 x ^{2}} + \frac{1}{2} (- \frac{e ^{x}}{x} + Ei (x))

Add a constant to the solution

= - \frac{e ^{x}}{2 x ^{2}} + \frac{1}{2} (- \frac{e ^{x}}{x} + Ei (x)) + C

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What is the integral of x^{-3}e^x ?
The integral of x^{-3}e^x is -(e^x}{2x^2}+1/2 (-\frac{e^x)/x+\Ei(x))+C