What is the integral of (e^{2x})/(25+e^{4x)} ?

The integral of (e^{2x})/(25+e^{4x)} is 1/10 arctan(1/5 e^{2x})+C

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integral of (e^{2x})/(25+e^{4x)}

\int \frac{e ^{2 x}}{25 + e ^{4 x}} d x

\frac{1}{10} arctan (\frac{1}{5} e^{2 x}) + C

Solution steps

\int \frac{e ^{2 x}}{25 + e ^{4 x}} d x

Apply u-substitution

: \int \frac{1}{2 ( 25 + u ^{2} )} d u

= \int \frac{1}{2 ( 25 + u ^{2} )} 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{1}{25 + u ^{2}} d u

Apply Integral Substitution

= \frac{1}{2} \cdot \int \frac{1}{5 ( v ^{2} + 1 )} 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 ^{2} + 1} d v

Use the common integral:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= \frac{1}{2} \cdot \frac{1}{5} arctan (v)

Substitute back

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

Simplify

\frac{1}{2} \cdot \frac{1}{5} arctan (\frac{e ^{2 x}}{5}) : \frac{1}{10} arctan (\frac{1}{5} e^{2 x})

= \frac{1}{10} arctan (\frac{1}{5} e^{2 x})

Add a constant to the solution

= \frac{1}{10} arctan (\frac{1}{5} e^{2 x}) + C

y = x^{2} 4 x - 7

f (x) = 6 sec (x), at x = \frac{π}{6}

n = 1 \sum \infty n e^{4 n}

x \to - 3 + lim (f (x))

f (x) = 3 x^{2} - x + 7, at x = 2

What is the integral of (e^{2x})/(25+e^{4x)} ?
The integral of (e^{2x})/(25+e^{4x)} is 1/10 arctan(1/5 e^{2x})+C