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

The integral of (e^{sqrt(2x)})/(sqrt(2x)) is e^{sqrt(2)sqrt(x)}+C

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

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

e^{2 x} + C

Solution steps

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

Simplify

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

= \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 2 e^{u} d u

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

Take the constant out:

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

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

Use the common integral:

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

= \frac{1}{2} 2 e^{u}

Substitute back

u = 2 x

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

Simplify

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

= e^{2 x}

Add a constant to the solution

= e^{2 x} + C

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