Primitive of Cotangent of a x over x

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Theorem

$\ds \int \frac {\cot a x} x \rd x = \frac {-1} a x - \frac {a x} 3 - \frac {\paren {a x}^3} {135} - \cdots - \frac {\paren {-1}^{n - 1} 2^{2 n} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} - \cdots + C$

where $B_n$ denotes the $n$th Bernoulli number.


Proof

\(\ds \int \frac {\csc a x} x \rd x\) \(=\) \(\ds \int \frac 1 x \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n}!} \rd x\) Power Series Expansion for Cosecant Function
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} a^{2 n - 1} } {\paren {2 n}!} \int x^{2 n - 2} \rd x\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} a^{2 n - 1} } {\paren {2 n}!} \paren {\frac {x^{2 n - 1} } {2 n - 1} } + C\) Primitive of Power
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} + C\)

$\blacksquare$


Also see


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