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

The integral of (e^{x^2})/(2x) is 1/4 \Ei(x^2)+C

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

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

\frac{1}{4} Ei (x^{2}) + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the nonelementary integral:

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

= \frac{1}{2} \cdot \frac{1}{2} Ei (u)

Substitute back

u = x^{2}

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

Simplify

\frac{1}{2} \cdot \frac{1}{2} Ei (x^{2}) : \frac{1}{4} Ei (x^{2})

= \frac{1}{4} Ei (x^{2})

Add a constant to the solution

= \frac{1}{4} Ei (x^{2}) + C

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What is the integral of (e^{x^2})/(2x) ?
The integral of (e^{x^2})/(2x) is 1/4 \Ei(x^2)+C