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

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

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

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

x^{2} - 1 - ln x^{2} - 1 + x + C

Solution steps

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

Expand

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

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

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

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

= x^{2} - 1 - ln x^{2} - 1 + x

Add a constant to the solution

= x^{2} - 1 - ln x^{2} - 1 + x + C

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