Primitive of x over Hyperbolic Cosine of a x minus 1

From ProofWiki
Jump to navigation Jump to search

Theorem

$\ds \int \frac {x \rd x} {\cosh a x - 1} = -\frac x a \coth \frac {a x} 2 + \frac 2 {a^2} \ln \size {\sinh \frac {a x} 2} + C$


Proof

With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

\(\ds u\) \(=\) \(\ds x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds 1\) Derivative of Identity Function


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \frac 1 {\cosh a x + 1}\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds -\frac 1 a \coth \frac {a x} 2\) Primitive of $\dfrac 1 {\cosh a x - 1}$


Then:

\(\ds \int \frac {x \rd x} {\cosh a x - 1}\) \(=\) \(\ds x \paren {-\frac 1 a \coth \frac {a x} 2} - \int \paren {-\frac 1 a \coth \frac {a x} 2} \rd x + C\) Integration by Parts
\(\ds \) \(=\) \(\ds -\frac x a \coth \frac {a x} 2 + \frac 1 a \int \coth {a x} 2 \rd x + C\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds -\frac x a \tanh \frac {a x} 2 + \frac 1 a \paren {\frac 2 a \ln \size {\sinh \frac {a x} 2} } + C\) Primitive of $\coth a x$
\(\ds \) \(=\) \(\ds -\frac x a \coth \frac {a x} 2 + \frac 2 {a^2} \ln \size {\sinh \frac {a x} 2} + C\) simplifying

$\blacksquare$


Also see


Sources