Primitive of Reciprocal of p squared plus Square of q by Hyperbolic Sine of a x
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Theorem
- $\ds \int \frac {\d x} {p^2 + q^2 \sinh^2 a x} = \begin {cases} \dfrac 1 {a p \sqrt{q^2 - p^2} } \arctan \dfrac {\sqrt {q^2 - p^2} \tanh a x} p + C & : p^2 < q^2 \\ \dfrac 1 {2 a p \sqrt{p^2 - q^2} } \ln \size {\dfrac {p + \sqrt {p^2 - q^2} \tanh a x} {p - \sqrt {p^2 - q^2} \tanh a x} } + C & : p^2 > q^2 \\ \end {cases}$
Proof
\(\ds \int \frac {\d x} {p^2 + q^2 \sinh^2 a x}\) | \(=\) | \(\ds \int \frac {\sech^2 a x \rd x} {p^2 \sech^2 a x + q^2 \tanh^2 a x}\) | multiplying numerator and denominator by $\sech^2 a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\sech^2 a x \rd x} {p^2 \paren {1 - \tanh^2 a x} + q^2 \tanh^2 a x}\) | Sum of Squares of Hyperbolic Secant and Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\sech^2 a x \rd x} {p^2 + \paren {q^2 - p^2} \tanh^2 a x}\) |
Then:
\(\ds u\) | \(=\) | \(\ds \tanh a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \d u\) | \(=\) | \(\ds a \sech^2 a x \rd x\) | Derivative of Hyperbolic Tangent Function | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\sech^2 a x \rd x} {p^2 + \paren {q^2 - p^2} \tanh^2 a x}\) | \(=\) | \(\ds \int \frac {\d u} {a p^2 + a \paren {q^2 - p^2} u^2}\) | Integration by Substitution: $u = \tanh a x$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a \paren {q^2 - p^2} } \int \frac {\d u} {u^2 + \paren {\frac {p^2} {q^2 - p^2} } }\) |
There are two cases to address: $q^2 - p^2 > 0$ and $q^2 - p^2 < 0$.
First suppose that $q^2 - p^2 > 0$.
Then:
\(\ds \frac 1 {a \paren {q^2 - p^2} } \int \frac {\d u} {u^2 + \paren {\frac {p^2} {q^2 - p^2} } }\) | \(=\) | \(\ds \frac 1 {a \paren {q^2 - p^2} } \int \frac {\d u} {u^2 + \paren {\frac p {\sqrt {q^2 - p^2} } }^2 }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a \paren {q^2 - p^2} } \frac {\sqrt {q^2 - p^2} } p \arctan \frac u {\frac p {\sqrt {q^2 - p^2} } } + C\) | Primitive of $\dfrac 1 {x^2 + a^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p \sqrt {q^2 - p^2} } \arctan \frac {\sqrt {q^2 - p^2} u} p + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a p \sqrt {q^2 - p^2} } \arctan \frac {\sqrt {q^2 - p^2} \tanh a x} p + C\) | substituting $u = \tanh a x$ |
Now suppose that $q^2 - p^2 < 0$.
Then:
\(\ds \frac 1 {a \paren {q^2 - p^2} } \int \frac {\d u} {u^2 + \paren {\frac {p^2} {q^2 - p^2} } }\) | \(=\) | \(\ds \frac 1 {a \paren {q^2 - p^2} } \int \frac {\d u} {u^2 - \paren {\frac {p^2} {p^2 - q^2} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-1} {a \paren {p^2 - q^2} } \int \frac {\d u} {u^2 - \paren {\frac p {\sqrt {p^2 - q^2} } }^2 }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-1} {a \paren {p^2 - q^2} } \frac {\sqrt {p^2 - q^2} } {2 p} \ln \size {\frac {u - \frac p {\sqrt {p^2 - q^2} } } {u + \frac p {\sqrt {p^2 - q^2} } } } + C\) | Primitive of $\dfrac 1 {x^2 - a^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-1} {2 a p \sqrt {p^2 - q^2} } \ln \size {\frac {\sqrt {p^2 - q^2} u - p} {\sqrt {p^2 - q^2} u + p} } + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a p \sqrt {p^2 - q^2} } \ln \size {\frac {\sqrt {p^2 - q^2} u + p} {\sqrt {p^2 - q^2} u - p} } + C\) | Logarithm of Reciprocal | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a p \sqrt {p^2 - q^2} } \ln \size {\frac {\sqrt {p^2 - q^2} \tanh a x + p} {\sqrt {p^2 - q^2} \tanh a x - p} } + C\) | substituting $u = \tanh a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a p \sqrt {p^2 - q^2} } \ln \size {\frac {p + \sqrt {p^2 - q^2} \tanh a x} {p - \sqrt {p^2 - q^2} \tanh a x} } + C\) | as $\size {a - b} - \size {b - a}$ |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x$: $14.555$