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

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