Primitive of Reciprocal of Power of Hyperbolic Cosine of a x
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Theorem
- $\ds \int \frac {\d x} {\cosh^n a x} = \frac {\sinh a x} {a \paren {n - 1} \cosh^{n - 1} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cosh^{n - 2} a x} + C$
Proof
| \(\ds \int \frac {\d x} {\cosh^n a x}\) | \(=\) | \(\ds \int \sech^n a x \rd x\) | Definition 2 of Hyperbolic Secant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sech^{n - 2} a x \tanh a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sech^{n - 2} a x \rd x + C\) | Primitive of $\sech^n a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\tanh a x} {a \paren {n - 1} \cosh^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cosh^{n - 2} a x} + C\) | Definition 2 of Hyperbolic Secant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sinh a x} {a \cosh a x \paren {n - 1} \cosh^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cosh^{n - 2} a x} + C\) | Definition 2 of Hyperbolic Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sinh a x} {a \paren {n - 1} \cosh^{n - 1} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cosh^{n - 2} a x} + C\) | simplifying |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cosh a x$: $14.588$
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 17$: Tables of Special Indefinite Integrals: $(28)$ Integrals Involving $\cosh a x$: $17.28.15.$