Primitive of Reciprocal of Hyperbolic Cosine of a x minus 1
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Theorem
- $\ds \int \frac {\d x} {\cosh a x - 1} = \frac {-1} a \coth \frac {a x} 2 + C$
Proof
\(\ds u\) | \(=\) | \(\ds \tanh \frac x 2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {\cosh x - 1}\) | \(=\) | \(\ds \int \frac {\dfrac {2 \rd u} {1 - u^2} } {\dfrac {1 + u^2} {1 - u^2} - 1}\) | Hyperbolic Tangent Half-Angle Substitution | ||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {2 \rd u} {1 + u^2 - \paren {1 - u^2} }\) | multiplying top and bottom by $1 - u^2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\d u} {u^2}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-1} u + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-1} {\tanh \dfrac x 2} + C\) | substituting for $u$ | |||||||||||
\(\ds \) | \(=\) | \(\ds -\coth \frac x 2 + C\) | Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {\cosh a x - 1}\) | \(=\) | \(\ds \frac {-1} a \coth \frac {a x} 2 + C\) | Primitive of Function of Constant Multiple |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cosh a x$: $14.576$