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

$\blacksquare$

Also see

• Primitive of $\dfrac 1 x \arcsin \dfrac x a$

• Primitive of $\dfrac 1 x \arccos \dfrac x a$

• Primitive of $\dfrac 1 x \arctan \dfrac x a$

• Primitive of $\dfrac 1 x \arccot \dfrac x a$

• Primitive of $\dfrac 1 x \arccsc \dfrac x a$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.496$