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

The integral of (e^x)/(3+5e^{2x)} is 1/(sqrt(15))arctan(sqrt(5/3)e^x)+C

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integral of (e^x)/(3+5e^{2x)}

\int \frac{e ^{x}}{3 + 5 e ^{2 x}} d x

\frac{1}{15} arctan (\frac{5}{3} e^{x}) + C

Solution steps

\int \frac{e ^{x}}{3 + 5 e ^{2 x}} d x

Apply u-substitution

: \int \frac{1}{3 + 5 u ^{2}} d u

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

Apply Integral Substitution

= \int \frac{1}{15 ( 1 + v ^{2} )} d v

Take the constant out:

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

= \frac{1}{15} \cdot \int \frac{1}{1 + v ^{2}} d v

Use the common integral:

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

= \frac{1}{15} arctan (v)

Substitute back

= \frac{1}{15} arctan (\frac{5}{3} e^{x})

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{1}{15} arctan (\frac{5}{3} e^{x})

Add a constant to the solution

= \frac{1}{15} arctan (\frac{5}{3} e^{x}) + C

\frac{d}{d t} (cos (π t))

\frac{\partial}{\partial x} (x^{3} - 3 x y^{2} + y^{3})

n = 1 \sum \infty \frac{7}{n} - \frac{7}{n + 1}

x \to 0 lim (- 6 + 6 x^{3})

n = 0 \sum \infty x^{n} \frac{1}{n !}

What is the integral of (e^x)/(3+5e^{2x)} ?
The integral of (e^x)/(3+5e^{2x)} is 1/(sqrt(15))arctan(sqrt(5/3)e^x)+C