Primitive of Reciprocal of 1 minus Cosine of a x/Corollary

Theorem

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

Proof

\(\ds \int \frac {\d x} {1 - \cos a x}\)

\(=\)

\(\ds -\frac 1 a \cot \frac {a x} 2 + C\)

Primitive of $\dfrac 1 {1 - \cos a x}$

\(\ds \leadsto \ \ \)

\(\ds \int \frac {\d x} {1 + \cos x}\)

\(=\)

\(\ds -\cot \frac x 2 + C\)

setting $a \gets 1$

$\blacksquare$

Sources

• 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals