What is the integral of 1/(x*e^x) ?

The integral of 1/(x*e^x) is \Ei(-x)+C

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integral of 1/(x*e^x)

\int \frac{1}{x \cdot e ^{x}} d x

Ei (- x) + C

Solution steps

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

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

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

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

Simplify

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

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

Apply u-substitution

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

= \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

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