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

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

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

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

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

Solution steps

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

Factor

e^{4 x} - 4 : - (- e^{4 x} + 4)

= \int \frac{e ^{2 x}}{- ( - e ^{4 x} + 4 )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply u-substitution

: \int - \frac{1}{4 u - u + 4} d u

= - \int - \frac{1}{4 u - u + 4} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply u-substitution

: \int - \frac{2}{- v ^{2} + 4} d v

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integral Substitution

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Use the common integral:

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

= - (- \frac{1}{4} (- 2 \cdot \frac{1}{2} (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2})))

Substitute back

= - - \frac{1}{4} - 2 \cdot \frac{1}{2} \frac{ln \frac{- ( - e ^{4 x} + 4 ) + 4}{2} + 1}{2} - \frac{ln \frac{- ( - e ^{4 x} + 4 ) + 4}{2} - 1}{2}

Simplify

- - \frac{1}{4} - 2 \cdot \frac{1}{2} \frac{ln \frac{- ( - e ^{4 x} + 4 ) + 4}{2} + 1}{2} - \frac{ln \frac{- ( - e ^{4 x} + 4 ) + 4}{2} - 1}{2} : - \frac{1}{8} (ln \frac{e ^{2 x}}{2} + 1 - ln \frac{e ^{2 x}}{2} - 1)

= - \frac{1}{8} (ln \frac{e ^{2 x}}{2} + 1 - ln \frac{e ^{2 x}}{2} - 1)

Add a constant to the solution

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

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