Primitive of Tangent of a x/Secant Form

Theorem

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

Proof

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

\(=\)

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

Primitive of $\tan x$: Secant Form

\(\ds \leadsto \ \ \)

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

\(=\)

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

Primitive of Function of Constant Multiple

\(\ds \)

\(=\)

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

simplifying

$\blacksquare$

Also see

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

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\tan a x$: $14.429$