Primitive of Reciprocal of Hyperbolic Cosine of a x

Theorem

$\ds \int \frac {\d x} {\cosh a x} = \frac {2 \map \arctan {e^{a x} } } a + C$

Proof

\(\ds \int \frac {\d x} {\cosh a x}\)

\(=\)

\(\ds \int \sech a x \rd x\)

Definition 2 of Hyperbolic Secant

\(\ds \)

\(=\)

\(\ds \frac {2 \map \arctan {e^{a x} } } a + C\)

Primitive of $\sech a x$: Arctangent of Exponential Form

$\blacksquare$

Also see

• Primitive of $\dfrac 1 {\sinh a x}$

• Primitive of $\dfrac 1 {\tanh a x}$

• Primitive of $\dfrac 1 {\coth a x}$

• Primitive of $\dfrac 1 {\sech a x}$

• Primitive of $\dfrac 1 {\csch a x}$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cosh a x$: $14.567$