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

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

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

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

π erfi (x) + C

Solution steps

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

Apply u-substitution

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

= \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

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

Use the nonelementary integral:

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

= 2 \cdot \frac{π}{2} erfi (u)

Substitute back

u = x

= 2 \cdot \frac{π}{2} erfi (x)

Simplify

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

= π erfi (x)

Add a constant to the solution

= π erfi (x) + C

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