Question

\[\int\frac{1}{1 - \cos x} dx\]

Answer 1

\[\int\frac{dx}{1 - \cos x}\]
\[ = \int\frac{dx}{1 - \cos x} \times \frac{1 + \ cosx}{1 + \ cosx}\]
\[ = \int\left( \frac{1 + \cos x}{1 - \cos^2 x} \right)dx\]
\[ = \int\left( \frac{1 + \cos x}{\sin^2 x} \right)dx\]
\[ = \int\left( \frac{1}{\sin^2 x} + \frac{\cos x}{\sin x} \times \frac{1}{\sin x} \right)dx\]
\[ = \int\left( {cosec}^2 x + \text{cosec x }\cot x \right)dx\]
\[ = - \cot x - \text{cosec x} + C\]