What is the integral of 1/x*e^{2x^2} ?

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

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integral of 1/x*e^{2x^2}

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

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

Solution steps

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

Simplify

\frac{1}{x} e^{2 x^{2}} : \frac{e ^{2 x^{2}}}{x}

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

Apply u-substitution

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

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

Apply u-substitution

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

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

Use the nonelementary integral:

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

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

Substitute back

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

Add a constant to the solution

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

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