Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Tangent Form
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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 $u$ |
$\blacksquare$
Also see
- Primitive of $\dfrac 1 {x^2 - a^2}$: $\coth^{-1}$ form for the case $\size z > a$
Sources
- 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Appendix $\text I$: Table of Indefinite Integrals $13$.