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

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

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integral of 2/(2-e^x)

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

x - ln ∣ e^{x} - 2 ∣ + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

= 2 \cdot \int \frac{1}{u ( 2 - u )} d u

Complete the square

u (2 - u) : - (u - 1)^{2} + 1

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

Apply u-substitution

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

Use the common integral:

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

= 2 (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

Substitute back

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

Simplify

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

= x - ln ∣ e^{x} - 2 ∣

Add a constant to the solution

= x - ln ∣ e^{x} - 2 ∣ + C