What is the integral of e^{3/x} ?

The integral of e^{3/x} is e^{3/x}x-3\Ei(3/x)+C

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integral of e^{3/x}

\int e^{\frac{3}{x}} d x

e^{\frac{3}{x}} x - 3 Ei (\frac{3}{x}) + C

Solution steps

\int e^{\frac{3}{x}} d x

Apply u-substitution

: \int - \frac{3 e ^{u}}{u ^{2}} d u

= \int - \frac{3 e ^{u}}{u ^{2}} d u

Take the constant out:

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

= - 3 \cdot \int \frac{e ^{u}}{u ^{2}} d u

Apply Integration By Parts

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

= - 3 (- \frac{e ^{u}}{u} - \int - \frac{e ^{u}}{u} d u)

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

= - 3 (- \frac{e ^{u}}{u} - (- Ei (u)))

Substitute back

u = \frac{3}{x}

= - 3 (- \frac{e ^{\frac{3}{x}}}{\frac{3}{x}} - (- Ei (\frac{3}{x})))

Simplify

- 3 (- \frac{e ^{\frac{3}{x}}}{\frac{3}{x}} - (- Ei (\frac{3}{x}))) : e^{\frac{3}{x}} x - 3 Ei (\frac{3}{x})

= e^{\frac{3}{x}} x - 3 Ei (\frac{3}{x})

Add a constant to the solution

= e^{\frac{3}{x}} x - 3 Ei (\frac{3}{x}) + C

\frac{s - 6}{3 s ^{2} + s + 12}

\int e^{\frac{x}{2} + 3} d x

\int \frac{2 y - 1}{y} d y

f (x) = e^{5}

\frac{d}{d x} (- \frac{128 x}{( 16 x ^{2} + 1 ) ^{2}})

What is the integral of e^{3/x} ?
The integral of e^{3/x} is e^{3/x}x-3\Ei(3/x)+C