Primitive of Reciprocal of p plus q by Hyperbolic Tangent of a x
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Theorem
- $\ds \int \frac {\d x} {p + q \tanh a x} = \frac {p x} {p^2 - q^2} - \frac q {a \paren {p^2 - q^2} } \ln \size {q \sinh a x + p \cosh a x} + C$
Proof
We have:
- $\dfrac \d {\d x} \paren {q \sinh a x + p \cosh a x} = a q \cosh a x + a p \sinh a x$
Thus:
\(\ds \int \frac {\d x} {p + q \tanh a x}\) | \(=\) | \(\ds \int \frac {\d x} {p + q \dfrac {\sinh a x} {\cosh a x} }\) | Definition 2 of Hyperbolic Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\cosh a x \rd x} {p \cosh a x + q \sinh a x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {p^2 - q^2} \int \frac {\paren {p^2 - q^2} \cosh a x \rd x} {p \cosh a x + q \sin a x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {p^2 - q^2} \int \frac {p^2 \cosh a x + p q \sinh a x - p q \sinh a x - q^2 \cosh a x} {p \cosh a x + q \sinh a x} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {p^2 - q^2} \paren {\int \frac {p^2 \cosh a x + p q \sinh a x} {p \cosh a x + q \sinh a x} \rd x - \int \frac {p q \sinh a x + q^2 \cosh a x} {p \cosh a x + q \sinh a x} \rd x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {p^2 - q^2} \paren {\int p \rd x - \frac q a \int \frac {\map \d {p \cosh a x + q \sinh a x} } {p \cosh a x + q \sinh a x} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {p x} {p^2 - q^2} - \frac q {a \paren {p^2 - q^2} } \ln \size {p \cosh a x + q \sinh a x} + C\) | Primitive of Constant and Primitive of Reciprocal |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\tanh a x$: $14.613$