Primitive of x over Sine of a x
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Theorem
\(\ds \int \frac {x \rd x} {\sin a x}\) | \(=\) | \(\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} {18} + \frac {7 \paren {a x}^5} {1800} \cdots} + C\) |
where $B_n$ denotes the $n$th Bernoulli number.
Proof
\(\ds \csc x\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \dfrac {B_{2 n} \paren {-1}^{n - 1} x^{2 n - 1} 2 \paren {2^{2 n - 1} - 1} } {\paren {2 n}!}\) | Power Series Expansion for Cosecant Function | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac x {\sin a x}\) | \(=\) | \(\ds x \sum_{n \mathop = 0}^\infty \dfrac {B_{2 n} \paren {-1}^{n - 1} \paren {a x}^{2 n - 1} 2 \paren {2^{2 n - 1} - 1} } {\paren {2 n}!}\) | Cosecant is Reciprocal of Sine | ||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 a \sum_{n \mathop = 0}^\infty \dfrac {B_{2 n} \paren {-1}^{n - 1} \paren {a x}^{2 n} 2 \paren {2^{2 n - 1} - 1} } {\paren {2 n}!}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \dfrac x {\sin a x} \rd x\) | \(=\) | \(\ds \dfrac 1 {a^2} \sum_{n \mathop = 0}^\infty \dfrac {B_{2 n} \paren {-1}^{n - 1} \paren {a x}^{2 n + 1} 2 \paren {2^{2 n - 1} - 1} } {\paren {2 n + 1}!}\) | Primitive of Power, Linear Combination of Primitives |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$: $14.346$