Primitive of Square of Hyperbolic Cotangent of a x

Theorem

$\ds \int \coth^2 a x \rd x = x - \frac {\coth a x} a + C$

Proof

\(\ds \int \coth^2 x \rd x\)

\(=\)

\(\ds x - \coth x + C\)

Primitive of $\coth^2 x$

\(\ds \leadsto \ \ \)

\(\ds \int \coth^2 a x \rd x\)

\(=\)

\(\ds \frac 1 a \paren {a x - \coth a x} + C\)

Primitive of Function of Constant Multiple

\(\ds \)

\(=\)

\(\ds x - \frac {\coth a x} a + C\)

simplifying

$\blacksquare$

Also see

• Primitive of $\sinh^2 a x$
• Primitive of $\cosh^2 a x$
• Primitive of $\tanh^2 a x$
• Primitive of $\sech^2 a x$
• Primitive of $\csch^2 a x$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\coth a x$: $14.616$
• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $126$.
• 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: $(31)$ Integrals Involving $\coth a x$: $17.31.2.$