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

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Theorem

Let $a \in \R_{>0}$ be a strictly positive real constant.

$\ds \int \dfrac {\d x} {x^2 - a^2} = \begin {cases} -\dfrac 1 a \tanh^{-1} \dfrac x a + C & : \size x < a \\

& \\ -\dfrac 1 a \coth^{-1} \dfrac x a + C & : \size x > a \\ & \\ \text {undefined} & : x = a \end {cases}$


Proof

First note that if $x = a$ then $a^2 - x^2 = 0$ and so $\dfrac 1 {x^2 - a^2}$ is undefined.


Proof for $\size x < a$

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 $u$

$\Box$


Proof for $\size x > a$

Let $\size x > a$.

Let:

\(\ds u\) \(=\) \(\ds \coth^{-1} {\frac x a}\) Definition of Real Inverse Hyperbolic Cotangent, which is defined where $\size {\dfrac x a} > 1$
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds a \coth u\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d x} {\d u}\) \(=\) \(\ds -a \csch^2 u\) Derivative of Hyperbolic Cotangent
\(\ds \leadsto \ \ \) \(\ds \int \frac 1 {x^2 - a^2} \rd x\) \(=\) \(\ds \int \frac {-a \csch^2 u} {a^2 \coth^2 u - a^2} \rd u\) Integration by Substitution
\(\ds \) \(=\) \(\ds -\frac a {a^2} \int \frac {\csch^2 u} {\coth^2 u - 1} \rd u\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds -\frac 1 a \int \frac {\csch^2 u} {\csch^2 u} \rd u\) Difference of Squares of Hyperbolic Cotangent and Cosecant
\(\ds \) \(=\) \(\ds -\frac 1 a \int \rd u\)
\(\ds \) \(=\) \(\ds -\frac 1 a u + C\) Integral of Constant
\(\ds \) \(=\) \(\ds -\frac 1 a \coth^{-1} {\frac x a} + C\) Definition of $u$

$\blacksquare$

Sources

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