Primitive of Cotangent of a x

Theorem

$\ds \int \cot a x \rd x = \frac {\ln \size {\sin a x} } a + C$

Proof

\(\ds \int \cot x \rd x\)

\(=\)

\(\ds \ln \size {\sin x}\)

Primitive of $\cot x$

\(\ds \leadsto \ \ \)

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

\(=\)

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

Primitive of Function of Constant Multiple

\(\ds \)

\(=\)

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

simplifying

$\blacksquare$

Examples

Cotangent of $x / 2$

$\ds \int \cot \dfrac x 2 \rd x = 2 \ln \size {\sin \dfrac 1 2} + C$

Also see

• Primitive of $\sin a x$
• Primitive of $\cos a x$
• Primitive of $\tan a x$
• Primitive of $\sec a x$
• Primitive of $\csc a x$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cot a x$: $14.440$
• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $83$.