What is the integral of (e^{2x})/(sqrt(1+5e^{2x))} ?

The integral of (e^{2x})/(sqrt(1+5e^{2x))} is 1/5 sqrt(1+5e^{2x)}+C

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integral of (e^{2x})/(sqrt(1+5e^{2x))}

\int \frac{e ^{2 x}}{1 + 5 e ^{2 x}} d x

\frac{1}{5} 1 + 5 e^{2 x} + C

Solution steps

\int \frac{e ^{2 x}}{1 + 5 e ^{2 x}} d x

Apply u-substitution

: \int \frac{e ^{u}}{2 1 + 5 e ^{u}} d u

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

Apply u-substitution

: \int \frac{1}{5 v} d v

= \frac{1}{2} \cdot \int \frac{1}{5 v} d v

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{5} \cdot \int \frac{1}{v} d v

Apply the Power Rule

: 2 v^{\frac{1}{2}}

= \frac{1}{2} \cdot \frac{1}{5} \cdot 2 v^{\frac{1}{2}}

Substitute back

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

Simplify

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

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

Add a constant to the solution

= \frac{1}{5} 1 + 5 e^{2 x} + C

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What is the integral of (e^{2x})/(sqrt(1+5e^{2x))} ?
The integral of (e^{2x})/(sqrt(1+5e^{2x))} is 1/5 sqrt(1+5e^{2x)}+C