What is the integral of (2-e^x)/(1+e^x) ?

The integral of (2-e^x)/(1+e^x) is 2x-3ln|e^x+1|+C

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integral of (2-e^x)/(1+e^x)

\int \frac{2 - e ^{x}}{1 + e ^{x}} d x

2 x - 3 ln ∣ e^{x} + 1 ∣ + C

Solution steps

\int \frac{2 - e ^{x}}{1 + e ^{x}} d x

Expand

\frac{2 - e ^{x}}{1 + e ^{x}} : \frac{2}{1 + e ^{x}} - \frac{e ^{x}}{1 + 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}{1 + e ^{x}} d x - \int \frac{e ^{x}}{1 + e ^{x}} d x

\int \frac{2}{1 + e ^{x}} d x = 2 (- ln ∣ 1 + e^{x} ∣ + x)

\int \frac{e ^{x}}{1 + e ^{x}} d x = ln ∣ 1 + e^{x} ∣

= 2 (- ln ∣ 1 + e^{x} ∣ + x) - ln ∣ 1 + e^{x} ∣

Simplify

2 (- ln ∣ 1 + e^{x} ∣ + x) - ln ∣ 1 + e^{x} ∣ : 2 x - 3 ln ∣ e^{x} + 1 ∣

= 2 x - 3 ln ∣ e^{x} + 1 ∣

Add a constant to the solution

= 2 x - 3 ln ∣ e^{x} + 1 ∣ + C

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What is the integral of (2-e^x)/(1+e^x) ?
The integral of (2-e^x)/(1+e^x) is 2x-3ln|e^x+1|+C