What is the integral of ((e^{sqrt(7x)}))/(sqrt(7x)) ?

The integral of ((e^{sqrt(7x)}))/(sqrt(7x)) is 2/7 e^{sqrt(7)sqrt(x)}+C

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integral of ((e^{sqrt(7x)}))/(sqrt(7x))

\int \frac{( e ^{7 x} )}{7 x} d x

\frac{2}{7} e^{7 x} + C

Solution steps

\int \frac{e ^{7 x}}{7 x} d x

Simplify

\frac{e ^{7 x}}{7 x} : \frac{e ^{7 x}}{7 x}

= \int \frac{e ^{7 x}}{7 x} d x

Take the constant out:

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

= \frac{1}{7} \cdot \int \frac{e ^{7 x}}{x} d x

Apply u-substitution

: \int \frac{2 e ^{u}}{7} d u

= \frac{1}{7} \cdot \int \frac{2 e ^{u}}{7} d u

Take the constant out:

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

= \frac{1}{7} \cdot \frac{2}{7} \cdot \int e^{u} d u

Use the common integral:

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

= \frac{1}{7} \cdot \frac{2}{7} e^{u}

Substitute back

u = 7 x

= \frac{1}{7} \cdot \frac{2}{7} e^{7 x}

Simplify

\frac{1}{7} \cdot \frac{2}{7} e^{7 x} : \frac{2}{7} e^{7 x}

= \frac{2}{7} e^{7 x}

Add a constant to the solution

= \frac{2}{7} e^{7 x} + C

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What is the integral of ((e^{sqrt(7x)}))/(sqrt(7x)) ?
The integral of ((e^{sqrt(7x)}))/(sqrt(7x)) is 2/7 e^{sqrt(7)sqrt(x)}+C