Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x minus 1
Jump to navigation
Jump to search
Theorem
- $\ds \int \frac {\d x} {\sinh a x \paren {\cosh a x - 1} } = \frac {-1} {2 a} \ln \size {\tanh \frac {a x} 2} - \frac 1 {2 a \paren {\cosh a x - 1} } + C$
Proof
\(\ds \int \frac {\d x} {\sinh a x \paren {\cosh a x - 1} }\) | \(=\) | \(\ds \int \frac {\map \sinh {a x} \rd x} {\sinh^2 a x \cosh a x - \sinh^2 a x}\) | multiplying through $\dfrac {\sinh a x} {\sinh a x}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\map \sinh {a x} \rd x} {\cosh a x \paren {\cosh^2 a x - 1} + 1 - \cosh^2 a x}\) | Difference of Squares of Hyperbolic Cosine and Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\map \sinh {a x} \rd x} {\cosh^3 a x - \cosh^2 a x - \cosh a x + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {\d t} {t^3 - t^2 - t + 1}\) | Integration by Substitution: $t \to \map \cosh {a x}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {4 a} \int \frac 1 {t + 1} \rd t - \frac 1 {4 a} \int \frac 1 {t - 1} \rd t + \frac 1 {2 a} \int \frac 1 {\paren {t - 1}^2} \rd t\) | Partial Fractions Expansion | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {4 a} \map \ln {t + 1} - \frac 1 {4 a} \map \ln {t - 1} - \frac 1 {2 a \paren {t - 1} } + C\) | Primitive of Reciprocal, Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {4 a} \map \ln { \frac {\cosh ax + 1} {\cosh ax - 1} } - \frac 1 {2 a \paren {\cosh a x - 1} } + C\) | substituting back $t \to \cosh ax$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {4 a} \map \ln {\frac {2 \cosh^2 \frac {a x} 2} {2 \sinh^2 \frac {a x} 2} } - \frac 1 {2 a \paren {\cosh ax - 1} } + C\) | Double Angle Formula for Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {4 a} \map \ln {\coth^2 \frac {a x} 2} - \frac 1 {2 a \paren {\cosh a x - 1} } + C\) | Definition of Hyperbolic Cotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \ln \size {\coth \frac {a x} 2} - \frac 1 {2 a \paren {\cosh a x - 1} } + C\) | Natural Logarithm of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 {2 a} \ln \size {\tanh \frac {a x} 2} - \frac 1 {2 a \paren {\cosh a x - 1} } + C\) | Logarithm of Reciprocal |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x $ and $\cosh a x$: $14.603$