Primitive of x by Cotangent of a x
Jump to navigation
Jump to search
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
- Primitive of $x \sin a x$
- Primitive of $x \cos a x$
- Primitive of $x \tan a x$
- Primitive of $x \sec a x$
- Primitive of $x \csc a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cot a x$: $14.446$