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

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

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integral of e^{1/(2x)}

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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