What is the integral of x^{-1}e^{-x} ?

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

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integral of x^{-1}e^{-x}

\int x^{- 1} e^{- x} d x

Ei (- x) + C

Solution steps

\int x^{- 1} e^{- x} d x

Simplify

x^{- 1} e^{- x} : \frac{e ^{- x}}{x}

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

Apply u-substitution

= \int \frac{e ^{u}}{u} d u

Use the nonelementary integral:

\int \frac{e ^{x}}{x} d x = Ei (x)

= Ei (u)

Substitute back

u = - x

= Ei (- x)

Add a constant to the solution

= Ei (- x) + C

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