Kugadzirisa chero nyaya yematekisi

Step by step

1. \int \frac{1}{x^{2} - 1}\, dx

   Find an antiderivative.

2. \frac{1}{x^{2} - 1} = \frac{- \frac{1}{x + 1} + \frac{1}{x - 1}}{2}

   Rewrite the integrand into a friendlier form.

3. \int \frac{- \frac{1}{x + 1} + \frac{1}{x - 1}}{2}\, dx = \frac{1}{2} \int - \frac{1}{x + 1} + \frac{1}{x - 1}\, dx

   Pull the constant \frac{1}{2} out of the integral.

4. \int - \frac{1}{x + 1} + \frac{1}{x - 1}\, dx = \int \frac{1}{x - 1}\, dx + \int - \frac{1}{x + 1}\, dx

   The integral of a sum is the sum of the integrals.

5. \int \frac{1}{x - 1}\, dx = \log{\left(x - 1 \right)}

   ∫ 1/u du = ln|u|.

6. \int - \frac{1}{x + 1}\, dx = -1 \int \frac{1}{x + 1}\, dx

   Pull the constant -1 out of the integral.

7. \int \frac{1}{x + 1}\, dx = \log{\left(x + 1 \right)}

   ∫ 1/u du = ln|u|.

8. F(x) = \frac{\log{\left(x - 1 \right)}}{2} - \frac{\log{\left(x + 1 \right)}}{2} + C

   Add the constant of integration.