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

1 Answer
Jun 18, 2018

int (1-2x)/(x+1)dx = -2 x+3 ln abs (x+1) +C12xx+1dx=2x+3ln|x+1|+C

Explanation:

Solve the indefinite integral:

int (1-2x)/(x+1)dx12xx+1dx

First reduce the degree of the numerator:

(1-2x)/(x+1) = - (2x-1)/(x+1) = -(2(x+1) -3)/(x+1) = -2+3/(x+1)12xx+1=2x1x+1=2(x+1)3x+1=2+3x+1

so, using the linearity of the integral:

int (1-2x)/(x+1)dx = -2 int dx +3 int dx/(x+1)12xx+1dx=2dx+3dxx+1

int (1-2x)/(x+1)dx = -2 x+3 ln abs (x+1) +C12xx+1dx=2x+3ln|x+1|+C