Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form
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Theorem
- $\ds \int \frac {\d x} {a^2 - x^2} = \frac 1 a \coth^{-1} \frac x a + C$
where $\size x > a$.
Proof
Let $\size x > a$.
Let:
| \(\ds u\) | \(=\) | \(\ds \coth^{-1} {\frac x a}\) | Definition of Real Inverse Hyperbolic Cotangent, which is defined where $\dfrac x a > 1$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds a \coth u\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {\d x} {\d u}\) | \(=\) | \(\ds -a \csch^2 u\) | Derivative of Hyperbolic Cotangent | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \int \frac {\rd x} {a^2 - x^2}\) | \(=\) | \(\ds \int \frac {-a \csch^2 u} {a^2 - a^2 \coth^2 u} \rd u\) | Integration by Substitution | ||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac a {a^2} \int \frac {\csch^2 u} {-\paren {\coth^2 u - 1} } \rd u\) | 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 $u$ |
Also see
- Primitive of $\dfrac 1 {a^2 - x^2}$: $\tanh^{-1}$ form for the case $\size x < a$
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $16$.