What is the integral from 0 to 1 of (e^x)/(sqrt(x)) ?

The integral from 0 to 1 of (e^x)/(sqrt(x)) is sqrt(pi)\erfi(1)

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integral from 0 to 1 of (e^x)/(sqrt(x))

\int_{0}^{1} \frac{e ^{x}}{x} d x

π erfi (1)

Solution steps

\int_{0}^{1} \frac{e ^{x}}{x} d x

Apply u-substitution

: \int_{0}^{1} 2 e^{u^{2}} d u

= \int_{0}^{1} 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_{0}^{1} e^{u^{2}} d u

Use the nonelementary integral:

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

= 2 [\frac{π}{2} erfi (u)]_{0}^{1}

Compute the boundaries:

\frac{π}{2} erfi (1)

= 2 \cdot \frac{π}{2} erfi (1)

Simplify

= π erfi (1)

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What is the integral from 0 to 1 of (e^x)/(sqrt(x)) ?
The integral from 0 to 1 of (e^x)/(sqrt(x)) is sqrt(pi)\erfi(1)