What is the integral of 1/(4e^{-2x)+e^{2x}} ?

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

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integral of 1/(4e^{-2x)+e^{2x}}

\int \frac{1}{4 e ^{- 2 x} + e ^{2 x}} d x

- \frac{1}{4} arctan (2 e^{- 2 x}) + C

Solution steps

\int \frac{1}{4 e ^{- 2 x} + e ^{2 x}} d x

Apply u-substitution

: \int - \frac{1}{2 ( 4 u ^{2} + 1 )} d u

= \int - \frac{1}{2 ( 4 u ^{2} + 1 )} 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}{4 u ^{2} + 1} d u

Apply Integral Substitution

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

Substitute back

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

Simplify

- \frac{1}{2} \cdot \frac{1}{2} arctan (2 e^{- 2 x}) : - \frac{1}{4} arctan (2 e^{- 2 x})

= - \frac{1}{4} arctan (2 e^{- 2 x})

Add a constant to the solution

= - \frac{1}{4} arctan (2 e^{- 2 x}) + C

\frac{d y}{d x} = \frac{( x ^{3} + 2 y ^{3} )}{x y ^{2}}

x^{(\frac{1}{2})}

x \to 0 + lim (2 x - 1)

\frac{d}{d x} (n ln (\frac{1}{x}))

\frac{\partial}{\partial x} (\frac{47 x ^{2}}{( x - y ) ^{2}})

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