Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x

Theorem

$\ds \int \frac {\d x} {\sinh a x \cosh a x} = \frac 1 a \ln \size {\tanh a x} + C$

Proof

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

\(=\)

\(\ds \int \frac {\sech a x \rd x} {\sinh a x}\)

Definition 2 of Hyperbolic Secant

\(\ds \)

\(=\)

\(\ds \int \frac {\sech^2 a x \rd x} {\sinh a x \sech a x}\)

multiplying top and bottom by $\sech a x$

\(\ds \)

\(=\)

\(\ds \int \frac {\sech^2 a x \rd x} {\frac {\sinh a x} {\cosh a x} }\)

Definition 2 of Hyperbolic Secant

\(\ds \)

\(=\)

\(\ds \int \frac {\sech^2 a x \rd x} {\tanh a x}\)

Definition 2 of Hyperbolic Tangent

\(\ds \)

\(=\)

\(\ds \frac 1 a \ln \size {\tanh a x} + C\)

Primitive of $\dfrac {\sech^2 a x} {\tanh a x}$

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x $ and $\cosh a x$: $14.595$