Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Tangent Form/Proof
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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 $u$ |
$\blacksquare$