Primitive of Reciprocal of 1 minus Cosine of a x

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Theorem

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


Proof

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

$\blacksquare$


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