Primitive of Function of Arcsecant

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$\ds \int \map F {\arcsec \frac x a} \rd x = a \int \map F u \sec u \tan u \rd u$

where $u = \arcsec \dfrac x a$.


First note that:

\(\ds u\) \(=\) \(\ds \arcsec \frac x a\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds a \sec u\) Definition of Arcsecant


\(\ds u\) \(=\) \(\ds \arcsec \frac x a\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac a {\size x {\sqrt {x^2 - a^2} } }\) Derivative of Arcsecant Function: Corollary 1
\(\ds \leadsto \ \ \) \(\ds \int \map F {\arcsec \frac x a} \rd x\) \(=\) \(\ds \int \map F u \, \frac {\size x {\sqrt {x^2 - a^2} } } a \rd u\) Primitive of Composite Function
\(\ds \) \(=\) \(\ds \int \map F u \frac {\size {a \sec u} {\sqrt {a^2 \sec^2 u - a^2} } } a \rd u\) Definition of $x$
\(\ds \) \(=\) \(\ds \int \map F u \size {\sec u} {\sqrt {a^2 \sec^2 u - a^2} } \rd u\)
\(\ds \) \(=\) \(\ds \int \map F u a \sec u {\sqrt {\sec^2 u - 1} } \rd u\) $\sec u > 0$ in this domain
\(\ds \) \(=\) \(\ds \int \map F u a \sec u \tan u \rd u\) Difference of Squares of Secant and Tangent
\(\ds \) \(=\) \(\ds a \int \map F u \sec u \tan u \rd u\) Primitive of Constant Multiple of Function


Also see