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

In this tutorial we shall discuss the integration of 1 over x into square root of x^2-1, and this is another important for 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 derivative, so we have

As we know that derivative of

This formula can also be written as