Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent Form

Theorem

$\ds \int \csch x \rd x = -2 \map {\coth^{-1} } {e^x} + C$

Proof

Let:

\(\ds \int \csch x \rd x\)

\(=\)

\(\ds \int \frac 2 {e^x - e^{-x} } \rd x\)

Definition of Hyperbolic Cosecant

\(\ds \)

\(=\)

\(\ds \int \frac {2 e^x} {e^{2 x} - 1} \rd x\)

multiplying top and bottom by $e^x$

Let:

\(\ds u\)

\(=\)

\(\ds e^x\)

\(\ds \leadsto \ \ \)

\(\ds u'\)

\(=\)

\(\ds e^x\)

Derivative of Exponential Function

Then:

\(\ds \int \csch x \rd x\)

\(=\)

\(\ds \int \frac {2 \rd u} {u^2 - 1}\)

Integration by Substitution

\(\ds \)

\(=\)

\(\ds -2 \coth^{-1} u + C\)

Primitive of Reciprocal of $x^2 - a^2$: $\coth^{-1}$ form

\(\ds \)

\(=\)

\(\ds -2 \map {\coth^{-1} } {e^x} + C\)

Definition of $u$

$\blacksquare$

Also see

• Primitive of Hyperbolic Secant Function

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.30$

(in which a mistake apppears)