# Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form

## Theorem

$\displaystyle \int \frac {\d x} {a^2 - x^2} = \frac 1 a \tanh^{-1} \frac x a + C$

where $x^2 < a^2$.

## Proof

Let:

 $\displaystyle \forall x \in \R: \size {\frac x a} < 1: \ \$ $\displaystyle u$ $=$ $\displaystyle \tanh^{-1} {\frac x a}$ $\displaystyle \forall x \in \R: x^2 < a^2: \ \$ $\displaystyle x$ $=$ $\displaystyle a \tanh u$ $\displaystyle \leadsto \ \$ $\displaystyle \forall x \in \R: x^2 < a^2: \ \$ $\displaystyle \frac {\d x} {\d u}$ $=$ $\displaystyle a \sech^2 u$ Derivative of Hyperbolic Tangent Function $\displaystyle \leadsto \ \$ $\displaystyle \forall x \in \R: x^2 < a^2: \ \$ $\displaystyle \int \frac 1 {a^2 - x^2} \rd x$ $=$ $\displaystyle \int \frac {a \sech^2 u} {a^2 - a^2 \tanh^2 u} \rd u$ Integration by Substitution $\displaystyle$ $=$ $\displaystyle \frac a {a^2} \int \frac {\sech^2 u} {1 - \tanh^2 u} \rd u$ Primitive of Constant Multiple of Function $\displaystyle$ $=$ $\displaystyle \frac 1 a \int \frac {\sech^2 u} {\sech^2 u} \rd u$ Sum of Squares of Hyperbolic Secant and Tangent $\displaystyle$ $=$ $\displaystyle \frac 1 a \int \rd u$ $\displaystyle$ $=$ $\displaystyle \frac 1 a u + C$ Integral of Constant $\displaystyle$ $=$ $\displaystyle \frac 1 a \tanh^{-1} \frac x a + C$ Definition of $u$

$\blacksquare$