What is the integral of 1/(4e^{-6x)+e^{6x}} ?

The integral of 1/(4e^{-6x)+e^{6x}} is -1/12 arctan(2e^{-6x})+C

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integral of 1/(4e^{-6x)+e^{6x}}

\int \frac{1}{4 e ^{- 6 x} + e ^{6 x}} d x

- \frac{1}{12} arctan (2 e^{- 6 x}) + C

Solution steps

\int \frac{1}{4 e ^{- 6 x} + e ^{6 x}} d x

Apply u-substitution

: \int - \frac{1}{6 ( 4 u ^{2} + 1 )} d u

= \int - \frac{1}{6 ( 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}{6} \cdot \int \frac{1}{4 u ^{2} + 1} d u

Apply Integral Substitution

= - \frac{1}{6} \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}{6} \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}{6} \cdot \frac{1}{2} arctan (v)

Substitute back

= - \frac{1}{6} \cdot \frac{1}{2} arctan (2 e^{- 6 x})

Simplify

- \frac{1}{6} \cdot \frac{1}{2} arctan (2 e^{- 6 x}) : - \frac{1}{12} arctan (2 e^{- 6 x})

= - \frac{1}{12} arctan (2 e^{- 6 x})

Add a constant to the solution

= - \frac{1}{12} arctan (2 e^{- 6 x}) + C

g (t) = \frac{3}{t ^{2}}

x \to 2 lim (5 x + 3)

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

\int cos^{4} (x) - cos^{6} (x) d x

\frac{\partial}{\partial y} (e^{x} + cos (y))

What is the integral of 1/(4e^{-6x)+e^{6x}} ?
The integral of 1/(4e^{-6x)+e^{6x}} is -1/12 arctan(2e^{-6x})+C