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

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

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integral of-ln(x^2-1)

\int - ln (x^{2} - 1) d x

- x ln (x^{2} - 1) - ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣ + 2 x + C

Solution steps

\int - ln (x^{2} - 1) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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