2. Calculate $\int \frac{\cos 2014 x}{\sin x} d x$.
Solution. Let $F_{n}=\int \frac{\cos n x}{\sin x} d x, n \in \mathbb{N}, x \in I \subset \mathbb{R}-\{k \pi / k \in \mathbb{Z}\}$

$$
F_{n+2}-F_{n}=\int \frac{\cos (n+2) x-\cos n x}{\sin x} d x=\int 2 \sin (n+1) x d x=\frac{2}{n+1} \cos (n+1) x+C
$$

$$
F_{0}=\ln \left|\operatorname{tg} \frac{x}{2}\right|+C
$$

$$
F_{2014}=\frac{2}{2013} \cos 2013 x+\frac{2}{2011} \cos 2011 x+\cdots+2 \cos x+\ln \left|\operatorname{tg} \frac{x}{2}\right|+C
$$

.3 points