Primitive of Secant of a x/Tangent Form

Theorem

$\ds \int \sec a x \rd x = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$

where $\map \tan {\dfrac \pi 4 + \dfrac {a x} 2} \ne 0$.

Proof

\(\ds \int \sec x \rd x\)

\(=\)

\(\ds \ln \size {\map \tan {\frac \pi 4 + \frac x 2} }\)

Primitive of $\sec x$: Tangent plus Angle Form

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C\)

Primitive of Function of Constant Multiple

$\blacksquare$

Also see

• Primitive of $\csc a x$: Tangent Form

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sec a x$: $14.451$