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

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Theorem

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

Let $\size x > a$.

Then:

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


Proof

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 $u$

$\blacksquare$


Also see


Sources

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