Primitive of Reciprocal of 1 minus Cosine of x

Theorem

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

Proof

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

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

The result follows by setting $a = 1$.

$\blacksquare$

Also see

• Primitive of $\dfrac 1 {1 + \cos x}$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals