Primitive of x by Cotangent of a x

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Theorem

$\ds \int x \cot a x \rd x = \frac 1 {a ^ 2} \paren {a x - \frac {\paren {a x}^3} 9 - \frac {\paren {a x}^5} {225} - \cdots + \frac {\paren {-1}^n 2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1} !} + \cdots} + C$

where $B_{2 n}$ denotes the $2 n$th Bernoulli number.


Proof

From Power Series Expansion for Cotangent Function:


The (real) cotangent function has a Taylor series expansion:

\(\ds \cot x\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}\)
\(\ds \) \(=\) \(\ds \frac 1 x - \frac x 3 - \frac {x^3} {45} - \frac {2 x^5} {945} + \frac {x^7} {4725} - \cdots\)


where $B_{2 n}$ denotes the Bernoulli numbers.

This converges for $0 < \size x < \pi$.


\(\ds x \cot ax\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} a^{2 n - 1} x^{2 n} } {\paren {2 n}!}\)
\(\ds \leadsto \ \ \) \(\ds \int x \cot a x \rd x\) \(=\) \(\ds \int \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} a^{2 n - 1} x^{2 n} } {\paren {2 n}!} \rd x\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {\int \frac {\paren {-1}^n 2^{2 n} B_{2 n} a^{2 n - 1} x^{2 n} } {\paren {2 n}!} \rd x}\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {\frac {\paren {-1}^n 2^{2 n} B_{2 n} a^{2 n - 1} } {\paren {2 n}!} \times \int x^{2 n} \rd x}\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {\frac {\paren {-1}^n 2^{2 n} B_{2 n} a^{2 n - 1} } {\paren {2 n}!} \times \frac {x^{2 n + 1} } {2 n + 1} + C}\) Primitive of Power
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} a^{2 n - 1} x^{2 n + 1} } {\paren {2 n + 1}!} + C\)
\(\ds \leadsto \ \ \) \(\ds \int x \cot a x \rd x\) \(=\) \(\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C\)

$\blacksquare$


Also see


Sources

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