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

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

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

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

- 2 x - 3 ln ∣ - x + 1 ∣ + 2 + C

Solution steps

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

Expand

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

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

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

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

Simplify

- ln ∣ 1 - x ∣ - 2 (- 1 + x + ln ∣ 1 - x ∣) : - 2 x - 3 ln ∣ - x + 1 ∣ + 2

= - 2 x - 3 ln ∣ - x + 1 ∣ + 2

Add a constant to the solution

= - 2 x - 3 ln ∣ - x + 1 ∣ + 2 + C

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