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

$\blacksquare$


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