Chapter 7 - Techniques of Integration - 7.5 Strategy for Integration - 7.5 Exercises - Page 548: 61

$$ -\cot x+\csc x+C $$

\begin{aligned} \int \frac{ d \theta}{ 1+\cos \theta } &= \int \frac{(1-\cos \theta) d \theta}{(1-\cos \theta)(1+\cos \theta)} \\ &=\int \frac{(1-\cos \theta) d \theta}{1-\cos ^{2} \theta}\\ &= \int \frac{(1-\cos \theta) d \theta}{\sin ^{2} \theta}\\ &= \int (\csc^2 x- \csc x\cot x)dx\\ &= -\cot x+\csc x+C \end{aligned}