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

## Theorem

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

where $\size x < a$.

## Proof

Let $\size x < a$.

Let:

 $\ds u$ $=$ $\ds \tanh^{-1} {\frac x a}$ Definition of Real Inverse Hyperbolic Tangent, which is defined where $\size {\dfrac x a} < 1$ $\ds \leadsto \ \$ $\ds x$ $=$ $\ds a \tanh u$ $\ds \leadsto \ \$ $\ds \frac {\d x} {\d u}$ $=$ $\ds a \sech^2 u$ Derivative of Hyperbolic Cotangent $\ds \leadsto \ \$ $\ds \int \frac 1 {x^2 - a^2} \rd x$ $=$ $\ds \int \frac {a \sech^2 u} {a^2 \tanh^2 u - a^2} \rd u$ Integration by Substitution $\ds$ $=$ $\ds \frac a {a^2} \int \frac {\sech^2 u} {-\paren {1 - \tanh^2 u} } \rd u$ Primitive of Constant Multiple of Function $\ds$ $=$ $\ds \frac 1 a \int \frac {\sech^2 u} {-\sech^2 u} \rd u$ Sum of Squares of Hyperbolic Secant and Tangent $\ds$ $=$ $\ds -\frac 1 a \int \rd u$ $\ds$ $=$ $\ds -\frac 1 a u + C$ Integral of Constant $\ds$ $=$ $\ds -\frac 1 a \tanh^{-1} {\frac x a} + C$ Definition of $u$

$\blacksquare$