Primitive of Reciprocal of 1 minus Cosine of a x

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Theorem

$\ds \int \frac {\d x} {1 - \cos a x} = -\frac 1 a \cot \frac {a x} 2 + C$


Corollary

$\ds \int \frac {\d x} {1 - \cos x} = -\cot \frac x 2 + C$


Proof

\(\ds u\) \(=\) \(\ds \tan \frac x 2\)
\(\ds \leadsto \ \ \) \(\ds \int \frac {\d x} {1 - \cos x}\) \(=\) \(\ds \int \frac {\dfrac {2 \rd u} {1 + u^2} } {1 - \dfrac {1 - u^2} {1 + u^2} }\) Weierstrass Substitution
\(\ds \) \(=\) \(\ds \int \frac {2 \rd u} {1 + u^2 - \paren {1 - u^2} }\) multiplying top and bottom by $1 + u^2$
\(\ds \) \(=\) \(\ds \int \frac {\d u} {u^2}\) simplifying
\(\ds \) \(=\) \(\ds \frac {-1} u + C\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac {-1} {\tan \dfrac x 2} + C\) substituting for $u$
\(\ds \) \(=\) \(\ds -\cot \frac x 2 + C\) Cotangent is Reciprocal of Tangent
\(\ds \leadsto \ \ \) \(\ds \int \frac {\d x} {1 - \cos a x}\) \(=\) \(\ds \frac {-1} a \cot \frac {a x} 2 + C\) Primitive of Function of Constant Multiple

$\blacksquare$


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