What is the integral of 1/(4e^{-7x)+e^{7x}} ?

The integral of 1/(4e^{-7x)+e^{7x}} is -1/14 arctan(2e^{-7x})+C

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integral of 1/(4e^{-7x)+e^{7x}}

\int \frac{1}{4 e ^{- 7 x} + e ^{7 x}} d x

- \frac{1}{14} arctan (2 e^{- 7 x}) + C

Solution steps

\int \frac{1}{4 e ^{- 7 x} + e ^{7 x}} d x

Apply u-substitution

: \int - \frac{1}{7 ( 4 u ^{2} + 1 )} d u

= \int - \frac{1}{7 ( 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}{7} \cdot \int \frac{1}{4 u ^{2} + 1} d u

Apply Integral Substitution

= - \frac{1}{7} \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}{7} \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}{7} \cdot \frac{1}{2} arctan (v)

Substitute back

= - \frac{1}{7} \cdot \frac{1}{2} arctan (2 e^{- 7 x})

Simplify

- \frac{1}{7} \cdot \frac{1}{2} arctan (2 e^{- 7 x}) : - \frac{1}{14} arctan (2 e^{- 7 x})

= - \frac{1}{14} arctan (2 e^{- 7 x})

Add a constant to the solution

= - \frac{1}{14} arctan (2 e^{- 7 x}) + C

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What is the integral of 1/(4e^{-7x)+e^{7x}} ?
The integral of 1/(4e^{-7x)+e^{7x}} is -1/14 arctan(2e^{-7x})+C