What is the integral of (e^x)/(2sqrt(x)) ?

The integral of (e^x)/(2sqrt(x)) is (sqrt(pi))/2 \erfi(sqrt(x))+C

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integral of (e^x)/(2sqrt(x))

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

\frac{π}{2} erfi (x) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int 2 e^{u^{2}} d u

= \frac{1}{2} \cdot \int 2 e^{u^{2}} d u

Take the constant out:

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

= \frac{1}{2} \cdot 2 \cdot \int e^{u^{2}} d u

Use the nonelementary integral:

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

= \frac{1}{2} \cdot 2 \cdot \frac{π}{2} erfi (u)

Substitute back

u = x

= \frac{1}{2} \cdot 2 \cdot \frac{π}{2} erfi (x)

Simplify

\frac{1}{2} \cdot 2 \cdot \frac{π}{2} erfi (x) : \frac{π}{2} erfi (x)

= \frac{π}{2} erfi (x)

Add a constant to the solution

= \frac{π}{2} erfi (x) + C

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What is the integral of (e^x)/(2sqrt(x)) ?
The integral of (e^x)/(2sqrt(x)) is (sqrt(pi))/2 \erfi(sqrt(x))+C