# Primitive of Arcsecant of x over a over x

## Theorem

 $\ds \int \dfrac 1 x \arcsec \frac x a \rd x$ $=$ $\ds \frac \pi 2 \ln \size x + \sum_{n \mathop \ge 0} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1}^2} \paren {\frac a x}^{2 n + 1} + C$ $\ds$ $=$ $\ds \frac \pi 2 \ln \size x + \frac a x + \frac 1 {2 \times 3 \times 3} \paren {\frac a x}^3 + \frac {1 \times 3} {2 \times 4 \times 5 \times 5} \paren {\frac a x}^5 + \frac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 7 \times 7} \paren {\frac a x}^7 + \cdots + C$

## Proof

 $\ds \arcsec \frac x a$ $=$ $\ds \frac \pi 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } \paren {\frac a x}^{2 n + 1}$ Power Series Expansion for Real Arcsecant Function $\ds \leadsto \ \$ $\ds \frac 1 x \arcsec \frac x a$ $=$ $\ds \frac \pi {2 x} - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } a^{2 n + 1} \paren {\frac 1 x}^{2 n + 2}$ $\ds \leadsto \ \$ $\ds \int \frac 1 x \arcsec \frac x a \rd x$ $=$ $\ds \int \paren {\frac \pi {2 x} - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } a^{2 n + 1} \paren {\frac 1 x}^{2 n + 2} } \rd x$ $\ds$ $=$ $\ds \frac \pi 2 \int \frac {\d x} x - \sum_{n \mathop = 0}^\infty \paren {\int \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } a^{2 n + 1} \paren {\frac 1 x}^{2 n + 2} \rd x}$ Fubini's Theorem $\ds$ $=$ $\ds \frac \pi 2 \ln \size x + \sum_{n \mathop = 0}^\infty {\paren {-\int \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } a^{2 n + 1} \paren {\frac 1 x}^{2 n + 2} \rd x} } + C$ Primitive of Reciprocal $\ds$ $=$ $\ds \frac \pi 2 \ln \size x + \sum_{n \mathop = 0}^\infty \frac {-\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} \paren {-\paren {2 n + 1} } } a^{2 n + 1} \paren {\frac 1 x}^{2 n + 1} + C$ Primitive of Power $\ds$ $=$ $\ds \frac \pi 2 \ln \size x + \sum_{n \mathop \ge 0} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1}^2} \paren {\frac a x}^{2 n + 1} + C$ rearranging

$\blacksquare$