Primitive of Secant of a x/Tangent Form
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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
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sec a x$: $14.451$