# Primitive of Reciprocal of x by Root of x squared minus a squared

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## Contents

## Theorem

- $\displaystyle \int \frac {\mathrm d x} {x \sqrt {x^2 - a^2} } = \frac 1 a \operatorname{arcsec} \left\vert{\frac x a}\right\vert + C$

## Proof

Let:

\(\displaystyle u\) | \(=\) | \(\displaystyle \operatorname{arcsec} {\frac x a}\) | |||||||||||

\(\displaystyle x\) | \(=\) | \(\displaystyle a \sec u\) | |||||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \frac {\mathrm d x} {\mathrm d u}\) | \(=\) | \(\displaystyle a \sec u \tan u\) | Derivative of Secant Function | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \int \frac {\mathrm d x} {x \sqrt {x^2 - a^2} }\) | \(=\) | \(\displaystyle \int \frac {a \sec u \tan u} {a \sec u \sqrt {a^2 \sec^2 u - a^2} } \ \mathrm d u\) | Integration by Substitution | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac a {a^2} \int \frac {\sec u \tan u} {\sec u \sqrt {\sec^2 u - 1} } \ \mathrm d u\) | Primitive of Constant Multiple of Function | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 a \int \frac {\sec u \tan u} {\sec u \tan u} \ \mathrm d u\) | Difference of Squares of Secant and Tangent | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 a \int 1 \ \mathrm d u\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 a u + C\) | Integral of Constant | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 a \operatorname{arcsec} {\frac x a} + C\) | Definition of $u$ |

$\blacksquare$

## Also see

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 14$: General Rules of Integration: $14.45$ - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {x^2 - a^2}$: $14.213$