Primitive of Arcsine of x over a over x

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Theorem

\(\displaystyle \int \frac {\arcsin \frac x a \ \mathrm d x} x\) \(=\) \(\displaystyle \sum_{k \mathop \ge 0} \frac {\left({2 k + 1}\right)!} {2^{2 k} \left({k!}\right)^2 \left({2 k + 1}\right)^3} \left({\frac x a}\right)^{2 k + 1}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac x a + \frac 1 {2 \times 3 \times 3} \left({\frac x a}\right)^3 + \frac {1 \times 3} {2 \times 4 \times 5 \times 5} \left({\frac x a}\right)^5 + \frac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 7 \times 7} \left({\frac x a}\right)^7 + \cdots + C\)


Proof


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Sources