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

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

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

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

ln x^{2} - 1 + \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ + C

Solution steps

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

Expand

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

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

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

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

Simplify

= ln x^{2} - 1 + \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣

Add a constant to the solution

= ln x^{2} - 1 + \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ + C

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