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

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

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

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

- 2 x + 3 ln ∣ x + 1 ∣ - 2 + C

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

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