# Integration of 1 Over (x Square Root of x^2-1)

In this tutorial we shall discuss the integration of 1 over x into the square root of x^2-1, and this is another important form of integration.

The integration of $\frac{1}{{x\sqrt {{x^2} - 1} }}$ is of the form

To prove this formula, consider

Using the derivative formula $\frac{d}{{dx}}{\sec ^{ - 1}}x = \frac{1}{{x\sqrt {{x^2} - 1} }}$, we have

Integrating both sides of equation (i) with respect to $x$, we have

As we know that by definition, integration is the inverse process of the derivative, so we have

We know that the derivative of

This formula can also be written as