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))

integral

\frac{( 1 + 2 x )}{( 1 - 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

5 e^{x}

\frac{( 1 - x )}{( x ^{2} - x + 4 )}

\frac{1}{v ( v + 1 )}

e^{x} e^{2 x} + 6 e^{x} + 10

1 - cos (2 x)

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