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

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

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integral of (2+e^x)/(2-e^x)

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

x - ln ∣ e^{x} - 2 ∣ - ln ∣ 2 - e^{x} ∣ + C

Solution steps

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

Expand

\frac{2 + e ^{x}}{2 - e ^{x}} : \frac{2}{2 - e ^{x}} + \frac{e ^{x}}{2 - e ^{x}}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{2}{2 - e ^{x}} d x = x - ln ∣ e^{x} - 2 ∣

\int \frac{e ^{x}}{2 - e ^{x}} d x = - ln ∣ 2 - e^{x} ∣

= x - ln ∣ e^{x} - 2 ∣ - ln ∣ 2 - e^{x} ∣

Add a constant to the solution

= x - ln ∣ e^{x} - 2 ∣ - ln ∣ 2 - e^{x} ∣ + C

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

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

y^{^{'}} = \frac{( 4 + 5 \cdot y )}{6}

\int x \cdot x^{\frac{1}{2}} d x

x \to 0 lim ((x^{2}) \cdot ln (x))

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