Primitive of Reciprocal of p squared minus Square of q by Hyperbolic Cosine of a x
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Theorem
- $\ds \int \frac {\d x} {p^2 - q^2 \cosh^2 a x} = \begin {cases}
\dfrac 1 {2 a p \sqrt {p^2 - q^2} } \ln \size {\dfrac {p \tanh a x + \sqrt {p^2 - q^2} } {p \tanh a x - \sqrt {p^2 - q^2} } } + C & : p^2 > q^2 \\ \dfrac 1 {a p \sqrt {q^2 - p^2} } \arctan \dfrac {p \tanh a x} {\sqrt {q^2 - p^2} } + C & : p^2 < q^2 \\ \end {cases}$
Proof
\(\ds \int \frac {\d x} {p^2 - q^2 \cosh^2 a x}\) | \(=\) | \(\ds \int \frac {\csch^2 a x \rd x} {p^2 \csch^2 a x - q^2 \coth^2 a x}\) | multiplying numerator and denominator by $\csch^2 a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\csch^2 a x \rd x} {p^2 \paren {\coth^2 a x - 1} - q^2 \coth^2 a x}\) | Difference of Squares of Hyperbolic Cotangent and Cosecant | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\csch^2 a x \rd x} {\paren {p^2 - q^2} \coth^2 a x - p^2}\) | simplifying |
Let:
\(\ds u\) | \(=\) | \(\ds \coth a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \d u\) | \(=\) | \(\ds -a \csch^2 a x \rd x\) | Derivative of Hyperbolic Cotangent Function | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {p^2 - q^2 \cosh^2 a x}\) | \(=\) | \(\ds \frac 1 a \int \frac {-\d u} {\paren {p^2 - q^2} u^2 - p^2}\) | Integration by Substitution: substituting $u = \tanh a x$ | ||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \frac 1 {a \paren {p^2 - q^2} } \int \frac {\rd u} {\frac {p^2} {p^2 - q^2} - u^2}\) | rearranging into a standard form |
There are two cases to address.
First, suppose $p^2 > q^2$.
Then we have that $p^2 - q^2 > 0$, and so:
\(\ds \int \frac {\d x} {p^2 - q^2 \cosh^2 a x}\) | \(=\) | \(\ds \frac 1 {a \paren {p^2 - q^2} } \int \frac {\rd u} {\paren {\frac p {\sqrt {p^2 - q^2} } }^2 - u^2}\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a \paren {p^2 - q^2} } \frac {\sqrt {p^2 - q^2} } {2 p} \ln \size {\frac {\frac p {\sqrt {p^2 - q^2} } + u} {\frac p {\sqrt {p^2 - q^2} } - u} } + C\) | Primitive of $\dfrac 1 {a^2 - x^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a p \sqrt {p^2 - q^2} } \ln \size {\frac {\frac p {\sqrt {p^2 - q^2} } + \coth a x} {\frac p {\sqrt {p^2 - q^2} } - \coth a x} } + C\) | substituting $u = \coth a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a p \sqrt {p^2 - q^2} } \ln \size {\frac {\frac p {\sqrt {p^2 - q^2} } + \frac 1 {\tanh a x} } {\frac p {\sqrt {p^2 - q^2} } - \frac 1 {\tanh a x} } } + C\) | Definition of Real Hyperbolic Cotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a p \sqrt {p^2 - q^2} } \ln \size {\frac {p \tanh a x + \sqrt {p^2 - q^2} } {p \tanh a x - \sqrt {p^2 - q^2} } } + C\) | multiplying numerator and denominator of argument of $\ln$ by $\sqrt {p^2 - q^2} \tanh a x$ |
Now suppose $p^2 < q^2$.
Then we have that $p^2 - q^2 < 0$, and so:
\(\ds \int \frac {\d x} {p^2 - q^2 \cosh^2 a x}\) | \(=\) | \(\ds \frac 1 {a \paren {p^2 - q^2} } \int \frac {\rd u} {\frac {p^2} {p^2 - q^2} - u^2}\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a \paren {p^2 - q^2} } \int \frac {\rd u} {\frac {-p^2} {q^2 - p^2} - u^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a \paren {q^2 - p^2} } \int \frac {\rd u} {u^2 + \paren {\frac p {\sqrt {q^2 - p^2} } }^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a \paren {q^2 - p^2} } \frac {\sqrt {q^2 - p^2} } p \arctan \frac u {\frac p {\sqrt {q^2 - p^2} } } + C\) | Primitive of $\dfrac 1 {x^2 + a^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p \sqrt {q^2 - p^2} } \arctan \frac {\sqrt {q^2 - p^2} u} p + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p \sqrt {q^2 - p^2} } \arctan \frac {\sqrt {q^2 - p^2} } p \coth a x + C\) | substituting $u = \coth a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p \sqrt {q^2 - p^2} } \arctan \frac {\sqrt {q^2 - p^2} {p \tanh a x} } + C\) | Definition of Real Hyperbolic Cotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p \sqrt {q^2 - p^2} } \arccot \frac {p \tanh a x} {\sqrt {q^2 - p^2} } + C\) | Arctangent of Reciprocal equals Arccotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p \sqrt {q^2 - p^2} } \map \arctan {\frac {p \tanh a x} {\sqrt {q^2 - p^2} } - \dfrac \pi 2} + C\) | Sum of Arctangent and Arccotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p \sqrt {q^2 - p^2} } \arctan \frac {p \tanh a x} {\sqrt {q^2 - p^2} } + C\) | subsuming $\dfrac \pi 2$ into constant |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cosh a x$: $14.583$