What is the integral of e^{x^{-1/2}} ?

The integral of e^{x^{-1/2}} is e^{1/(x^{1/2)}}x+e^{1/(x^{1/2)}}x^{1/2}-\Ei(x^{-1/2})+C

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integral of e^{x^{-1/2}}

\int e^{x^{- \frac{1}{2}}} d x

e^{\frac{1}{x ^{\frac{1}{2}}}} x + e^{\frac{1}{x ^{\frac{1}{2}}}} x^{\frac{1}{2}} - Ei (x^{- \frac{1}{2}}) + C

Solution steps

\int e^{x^{- \frac{1}{2}}} d x

Apply u-substitution

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

Take the constant out:

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

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

Apply Integration By Parts

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

\int - \frac{e ^{u}}{2 u ^{2}} d u = - \frac{1}{2} (- \frac{e ^{u}}{u} + Ei (u))

= - 2 (- \frac{e ^{u}}{2 u ^{2}} - (- \frac{1}{2} (- \frac{e ^{u}}{u} + Ei (u))))

Substitute back

u = x^{- \frac{1}{2}}

= - 2 - \frac{e ^{x^{- \frac{1}{2}}}}{2 ( x ^{- \frac{1}{2}} ) ^{2}} - (- \frac{1}{2} (- \frac{e ^{x^{- \frac{1}{2}}}}{x ^{- \frac{1}{2}}} + Ei (x^{- \frac{1}{2}})))

Simplify

- 2 - \frac{e ^{x^{- \frac{1}{2}}}}{2 ( x ^{- \frac{1}{2}} ) ^{2}} - (- \frac{1}{2} (- \frac{e ^{x^{- \frac{1}{2}}}}{x ^{- \frac{1}{2}}} + Ei (x^{- \frac{1}{2}}))) : e^{\frac{1}{x ^{\frac{1}{2}}}} x + e^{\frac{1}{x ^{\frac{1}{2}}}} x^{\frac{1}{2}} - Ei (x^{- \frac{1}{2}})

= e^{\frac{1}{x ^{\frac{1}{2}}}} x + e^{\frac{1}{x ^{\frac{1}{2}}}} x^{\frac{1}{2}} - Ei (x^{- \frac{1}{2}})

Add a constant to the solution

= e^{\frac{1}{x ^{\frac{1}{2}}}} x + e^{\frac{1}{x ^{\frac{1}{2}}}} x^{\frac{1}{2}} - Ei (x^{- \frac{1}{2}}) + C

What is the integral of e^{x^{-1/2}} ?
The integral of e^{x^{-1/2}} is e^{1/(x^{1/2)}}x+e^{1/(x^{1/2)}}x^{1/2}-\Ei(x^{-1/2})+C