Primitive of Sine of a x over Cosine of a x

Theorem

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

Proof

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

\(=\)

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

Tangent is Sine divided by Cosine

\(\ds \)

\(=\)

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

Primitive of $\tan a x$: Cosine Form

$\blacksquare$

Also see

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

Sources

• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $67$.