What is the integral of (e^{sqrt(5x)})/(sqrt(3x)) ?

The integral of (e^{sqrt(5x)})/(sqrt(3x)) is 2/(sqrt(15))e^{sqrt(5)sqrt(x)}+C

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integral of (e^{sqrt(5x)})/(sqrt(3x))

\int \frac{e ^{5 x}}{3 x} d x

\frac{2}{15} e^{5 x} + C

Solution steps

\int \frac{e ^{5 x}}{3 x} d x

Simplify

\frac{e ^{5 x}}{3 x} : \frac{e ^{5 x}}{3 x}

= \int \frac{e ^{5 x}}{3 x} d x

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{e ^{5 x}}{x} d x

Apply u-substitution

= \frac{1}{3} \cdot \int \frac{2 e ^{u}}{5} d u

Take the constant out:

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

= \frac{1}{3} \cdot \frac{2}{5} \cdot \int e^{u} d u

Use the common integral:

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

= \frac{1}{3} \cdot \frac{2}{5} e^{u}

Substitute back

u = 5 x

= \frac{1}{3} \cdot \frac{2}{5} e^{5 x}

Simplify

\frac{1}{3} \cdot \frac{2}{5} e^{5 x} : \frac{2}{15} e^{5 x}

= \frac{2}{15} e^{5 x}

Add a constant to the solution

= \frac{2}{15} e^{5 x} + C

What is the integral of (e^{sqrt(5x)})/(sqrt(3x)) ?
The integral of (e^{sqrt(5x)})/(sqrt(3x)) is 2/(sqrt(15))e^{sqrt(5)sqrt(x)}+C