What is the integral of (e^x)/(1+25e^{2x)} ?

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

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integral of (e^x)/(1+25e^{2x)}

\int \frac{e ^{x}}{1 + 25 e ^{2 x}} d x

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

Solution steps

\int \frac{e ^{x}}{1 + 25 e ^{2 x}} d x

Apply u-substitution

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

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

Apply Integral Substitution

= \int \frac{1}{5 ( 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}{5} \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}{5} arctan (v)

Substitute back

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

Add a constant to the solution

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

f (x) = 2 x + \frac{3}{x}, at x = - 1

\frac{\partial}{\partial x} (\frac{arctan ( \cdot y ^{2} )}{x})

\frac{d y}{d t} + e^{t} y = - e^{t}

2 y^{^{''}} + 3 y^{^{'}} + 9 y = 0

\frac{( x ^{4} - 5 x ^{3} + x )}{x ^{2}}

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