What is the integral of (e^{y/x})/x ?

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

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integral of (e^{y/x})/x

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

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

Solution steps

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

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

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

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

Add a constant to the solution

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

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