Primitive of x over Hyperbolic Cosine of a x minus 1
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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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cosh a x$: $14.578$