Primitive of Tangent of a x/Cosine Form

Theorem

$\ds \int \tan a x \rd x = \frac {-\ln \size {\cos a x} } a + C$

Proof

\(\ds \int \tan x \rd x\)

\(=\)

\(\ds -\ln \size {\cos x}\)

Primitive of $\tan x$: Cosine Form

\(\ds \leadsto \ \ \)

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

\(=\)

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

Primitive of Function of Constant Multiple

\(\ds \)

\(=\)

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

simplifying

$\blacksquare$

Also see

• Primitive of $\tan a x$: Secant Form

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\tan a x$: $14.429$
• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $82$.