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
- 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals