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

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

where $\size x < a$.


Let $\size x < a$.


\(\ds u\) \(=\) \(\ds \tanh^{-1} {\frac x a}\) 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 $u$


Also see