What is the integral of (3x-4)/(x^2-1) ?

The integral of (3x-4)/(x^2-1) is 3/2 ln|x^2-1|+4(1/2 ln|x+1|-1/2 ln|x-1|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (3x-4)/(x^2-1)

\int \frac{3 x - 4}{x ^{2} - 1} d x

\frac{3}{2} ln x^{2} - 1 + 4 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣) + C

Solution steps

\int \frac{3 x - 4}{x ^{2} - 1} d x

Expand

\frac{3 x - 4}{x ^{2} - 1} : \frac{3 x}{x ^{2} - 1} - \frac{4}{x ^{2} - 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{3 x}{x ^{2} - 1} d x - \int \frac{4}{x ^{2} - 1} d x

\int \frac{3 x}{x ^{2} - 1} d x = \frac{3}{2} ln x^{2} - 1

\int \frac{4}{x ^{2} - 1} d x = - 4 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣)

= \frac{3}{2} ln x^{2} - 1 - (- 4 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣))

Simplify

= \frac{3}{2} ln x^{2} - 1 + 4 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣)

Add a constant to the solution

= \frac{3}{2} ln x^{2} - 1 + 4 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣) + C

What is the integral of (3x-4)/(x^2-1) ?
The integral of (3x-4)/(x^2-1) is 3/2 ln|x^2-1|+4(1/2 ln|x+1|-1/2 ln|x-1|)+C