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

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

The integration of $\frac{1}{{\sqrt {{x^2} – {a^2}} }}$ is of the form
$\int {\frac{1}{{\sqrt {{x^2} – {a^2}} }}dx = } {\cosh ^{ – 1}}\left( {\frac{x}{a}} \right) + c$

To prove this formula, putting $x = a\cosh t$ we have $dx = a\sinh tdt$, $t = {\cosh ^{ – 1}}\left( {\frac{x}{a}} \right)$. So the given integral takes the form
$\begin{gathered} \int {\frac{{dx}}{{\sqrt {{x^2} – {a^2}} }} = \int {\frac{{a\sinh tdt}}{{\sqrt {{a^2}{{\cosh }^2}t – {a^2}} }}} } \\ \Rightarrow \int {\frac{{dx}}{{\sqrt {{x^2} – {a^2}} }} = \int {\frac{{a\sinh tdt}}{{a\sqrt {{{\cosh }^2}t – 1} }}} } \\ \Rightarrow \int {\frac{{dx}}{{\sqrt {{x^2} – {a^2}} }} = \int {\frac{{\sinh tdt}}{{\sqrt {{{\sinh }^2}t} }}} } = \int {dt} \\ \Rightarrow \int {\frac{{dx}}{{\sqrt {{x^2} – {a^2}} }} = t + c} \\ \end{gathered}$

Using the value $t = {\cosh ^{ – 1}}\left( {\frac{x}{a}} \right)$, we have
$\Rightarrow \int {\frac{{dx}}{{\sqrt {{x^2} – {a^2}} }} = {{\cosh }^{ – 1}}\left( {\frac{x}{a}} \right) + c}$