Primitive of Cosine of a x over Sine of a x

Theorem

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

Proof

\(\ds \int \dfrac {\cos a x} {\sin a x} \rd x\)

\(=\)

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

Cotangent is Cosine divided by Sine

\(\ds \)

\(=\)

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

Primitive of $\cot a x$

$\blacksquare$

Also see

• Primitive of $\dfrac {\sin a x} {\cos a x}$

Sources

• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $65$.