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

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

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

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

e^{a x} \frac{2}{a} + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

= 2 \cdot \int e^{a u} d u

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

= 2 \cdot \frac{1}{a} e^{v}

Substitute back

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

Simplify

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

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

Add a constant to the solution

= e^{a x} \frac{2}{a} + C

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