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

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

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

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

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

Solution steps

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

Simplify

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

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

Apply u-substitution

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

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

= - Ei (\frac{1}{x})

Add a constant to the solution

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

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