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

The integral of (2x+2)/(1-x) is 2-2x-4ln|1-x|+C

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

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

2 - 2 x - 4 ln ∣ 1 - x ∣ + C

Solution steps

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

Expand

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

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

\int \frac{2}{1 - x} d x = - 2 ln ∣ 1 - x ∣

= - 2 (- 1 + x + ln ∣ 1 - x ∣) - 2 ln ∣ 1 - x ∣

Simplify

- 2 (- 1 + x + ln ∣ 1 - x ∣) - 2 ln ∣ 1 - x ∣ : 2 - 2 x - 4 ln ∣ 1 - x ∣

= 2 - 2 x - 4 ln ∣ 1 - x ∣

Add a constant to the solution

= 2 - 2 x - 4 ln ∣ 1 - x ∣ + C

What is the integral of (2x+2)/(1-x) ?
The integral of (2x+2)/(1-x) is 2-2x-4ln|1-x|+C