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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int - \frac{1}{2 u} d u

= \int - \frac{1}{2 u} 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}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{2} ln ∣ u ∣

Substitute back

u = 2 - e^{2 x}

= - \frac{1}{2} ln 2 - e^{2 x}

Add a constant to the solution

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

y^{^{''}} - \frac{1}{t} y^{^{'}} + \frac{2}{t ^{2}} y = 0

\int x 4 - 3 x^{2} d x

\frac{\partial}{\partial x} (x e^{\frac{y}{z}})

\frac{\partial}{\partial x} (arcsin (\frac{x}{y ^{2}}))

\frac{d}{d x} (A x e^{- x})

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