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

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

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integral of e^{a/x}

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

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

Solution steps

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

Apply u-substitution

: \int - \frac{a e ^{u}}{u ^{2}} d u

= \int - \frac{a e ^{u}}{u ^{2}} d u

Take the constant out:

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

= - a \cdot \int \frac{e ^{u}}{u ^{2}} d u

Apply Integration By Parts

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

= - a (- \frac{e ^{u}}{u} - \int - \frac{e ^{u}}{u} d u)

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

= - a (- \frac{e ^{u}}{u} - (- Ei (u)))

Substitute back

u = \frac{a}{x}

= - a (- \frac{e ^{\frac{a}{x}}}{\frac{a}{x}} - (- Ei (\frac{a}{x})))

Simplify

- a (- \frac{e ^{\frac{a}{x}}}{\frac{a}{x}} - (- Ei (\frac{a}{x}))) : e^{\frac{a}{x}} x - a Ei (\frac{a}{x})

= e^{\frac{a}{x}} x - a Ei (\frac{a}{x})

Add a constant to the solution

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

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