The Notes: `int dx/(cos(x) - 1)` Evaluate the integral

$\displaystyle\int\frac{{\left.{d}{x}}}{{{\cos{{x}}}-{1}}}=$

Multiply by the conjugate of the denominator.

$\displaystyle\int\frac{{1}}{{{\cos{{x}}}-{1}}}\cdot\frac{{{\cos{{x}}}+{1}}}{{{\cos{{x}}}+{1}}}{\left.{d}{x}\right.}=$

$\displaystyle\int\frac{{{\cos{{x}}}+{1}}}{{{{\cos}}^{{2}}{x}-{1}}}{\left.{d}{x}\right.}=$

Simplify, then integrate.

$\displaystyle\int\frac{{{\cos{{x}}}+{1}}}{{-{{\sin}}^{{2}}{x}}}{\left.{d}{x}\right.}=$

$\displaystyle\int\frac{{\cos{{x}}}}{{-{{\sin}}^{{2}}{x}}}{\left.{d}{x}\right.}+\int\frac{{1}}{{-{{\sin}}^{{2}}{x}}}{\left.{d}{x}\right.}=$

$\displaystyle\int\frac{{\cos{{x}}}}{{-{\sin{{x}}}{\sin{{x}}}}}{\left.{d}{x}\right.}+\int{\left(-{{\csc}}^{{2}}{x}\right)}{\left.{d}{x}\right.}=$

$\displaystyle\int{\left(-{\cot{{x}}}{\csc{{x}}}-{{\csc}}^{{2}}{x}\right)}{\left.{d}{x}\right.}=$

$\displaystyle{\csc{{x}}}+{\cot{{x}}}+{C}$

The Notes