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

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

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

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

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

Solution steps

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

Apply u-substitution

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

= \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} Ei (u)

Substitute back

u = 2 x

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

Add a constant to the solution

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

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