Primitive of x by Cosecant of a x
Jump to navigation
Jump to search
Theorem
- $\ds \int x \csc a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} {18} + \frac {7 \paren {a x}^5} {1800} + \cdots + \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
where $B_{2 n}$ is the $2 n$th Bernoulli number.
Proof
\(\ds \int x \csc a x \rd x\) | \(=\) | \(\ds \frac 1 {a^2} \int \theta \csc \theta \rd \theta\) | Substitution of $a x \to \theta$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} \rd \theta\) | Power Series Expansion for Cosecant Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} } {\paren {2 n}!} \int \theta^{2 n} \rd \theta\) | Power Series is Termwise Integrable within Radius of Convergence | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C\) | Substituting back $\theta \to ax$ |
$\blacksquare$
Also see
- Primitive of $x \sin a x$
- Primitive of $x \cos a x$
- Primitive of $x \tan a x$
- Primitive of $x \cot a x$
- Primitive of $x \sec a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\csc a x$: $14.466$