Primitive of Arcsine of x over a over x

Theorem

$\ds \int \frac 1 x \arcsin \frac x a \rd x$

$=$

$\ds \sum_{n \mathop \ge 0} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1}^2} \paren {\frac x a}^{2 n + 1}$

$\ds $

$=$

$\ds \frac x a + \frac 1 {2 \times 3 \times 3} \paren {\frac x a}^3 + \frac {1 \times 3} {2 \times 4 \times 5 \times 5} \paren {\frac x a}^5 + \frac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 7 \times 7} \paren {\frac x a}^7 + \cdots + C$

Proof

$\ds \arcsin \frac x a$

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } \paren {\frac x a}^{2 n + 1}$

Power Series Expansion for Real Arcsine Function

$\ds \leadsto \ \ $

$\ds \frac 1 x \arcsin \frac x a$

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } \paren {\frac 1 a}^{2 n + 1} x^{2 n}$

$\ds \leadsto \ \ $

$\ds \int \frac 1 x \arcsin \frac x a \rd x$

$=$

$\ds \int \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } \paren {\frac 1 a}^{2 n + 1} x^{2 n} } \rd x$

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \paren {\int {\frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } \paren {\frac 1 a}^{2 n + 1} x^{2 n} } \rd x}$

Power Series is Termwise Integrable within Radius of Convergence

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} } \paren {\frac 1 a}^{2 n + 1} \frac {x^{2 n + 1} } {2 n + 1}$

Primitive of Power

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1}^2} \paren {\frac x a}^{2 n + 1}$

rearranging

$\blacksquare$