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

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

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

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

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

Solution steps

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

Simplify

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

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{2 e ^{u}}{5} d u

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

Use the common integral:

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

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

Substitute back

u = 5 x

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

Simplify

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

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

Add a constant to the solution

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

f (x) = 12 x - 2 x, at x = 9

\int \frac{3}{9 + x ^{2}} d x

e^{s i n (2 x)}

\int - 2 cos (x) + 3 x^{2} - 5 x + 4 d x

\int 3 x + \frac{1}{3 x} d x

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