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

## 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 $u$

$\blacksquare$