What is the integral of 1/(4e^{-8x)+e^{8x}} ?

The integral of 1/(4e^{-8x)+e^{8x}} is -1/16 arctan(2e^{-8x})+C

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integral of 1/(4e^{-8x)+e^{8x}}

\int \frac{1}{4 e ^{- 8 x} + e ^{8 x}} d x

- \frac{1}{16} arctan (2 e^{- 8 x}) + C

Solution steps

\int \frac{1}{4 e ^{- 8 x} + e ^{8 x}} d x

Apply u-substitution

: \int - \frac{1}{8 ( 4 u ^{2} + 1 )} d u

= \int - \frac{1}{8 ( 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}{8} \cdot \int \frac{1}{4 u ^{2} + 1} d u

Apply Integral Substitution

= - \frac{1}{8} \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}{8} \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}{8} \cdot \frac{1}{2} arctan (v)

Substitute back

= - \frac{1}{8} \cdot \frac{1}{2} arctan (2 e^{- 8 x})

Simplify

- \frac{1}{8} \cdot \frac{1}{2} arctan (2 e^{- 8 x}) : - \frac{1}{16} arctan (2 e^{- 8 x})

= - \frac{1}{16} arctan (2 e^{- 8 x})

Add a constant to the solution

= - \frac{1}{16} arctan (2 e^{- 8 x}) + C

\frac{\partial}{\partial x} (x (x^{2} + y^{2} + z^{2} - 1))

x \to 1 - lim (x^{2} - 5)

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

\frac{d}{d x} y - y = 2

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

What is the integral of 1/(4e^{-8x)+e^{8x}} ?
The integral of 1/(4e^{-8x)+e^{8x}} is -1/16 arctan(2e^{-8x})+C