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

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

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