# Primitive of Reciprocal of x by Root of a squared minus x squared/Inverse Hyperbolic Secant Form

## Theorem

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

## Proof

Let:

 $\displaystyle u$ $=$ $\displaystyle \operatorname{sech}^{-1} {\frac x a}$ $\displaystyle x$ $=$ $\displaystyle a \operatorname{sech} u$ $\displaystyle \implies \ \$ $\displaystyle \frac {\mathrm d x} {\mathrm d u}$ $=$ $\displaystyle -a \operatorname{sech} u \tanh u$ Derivative of Hyperbolic Secant Function $\displaystyle \implies \ \$ $\displaystyle \int \frac {\mathrm d x} {x \sqrt {a^2 - x^2} }$ $=$ $\displaystyle \int \frac {-a \operatorname{sech} u \tanh u} {a \operatorname{sech} u \sqrt {a^2 - a^2 \operatorname{sech}^2 u} } \ \mathrm d u$ Integration by Substitution $\displaystyle$ $=$ $\displaystyle -\frac a {a^2} \int \frac {\operatorname{sech} u \tanh u} {\operatorname{sech} u \sqrt {1 - \operatorname{sech}^2 u} } \ \mathrm d u$ Primitive of Constant Multiple of Function $\displaystyle$ $=$ $\displaystyle -\frac 1 a \int \frac {\operatorname{sech} u \tanh u} {\operatorname{sech} u \tanh u} \ \mathrm d u$ Sum of Squares of Hyperbolic Secant and Tangent $\displaystyle$ $=$ $\displaystyle -\frac 1 a \int 1 \ \mathrm d u$ $\displaystyle$ $=$ $\displaystyle -\frac 1 a u + C$ Integral of Constant $\displaystyle$ $=$ $\displaystyle -\frac 1 a \operatorname{sech}^{-1} {\frac x a} + C$ Definition of $u$

$\blacksquare$