<s>[INST] How to find $\int \frac{\sin x}{2\cos 2x} \operatorname{d}x$? How to compute $$ \int \frac{\sin x}{2\cos 2x} \operatorname{d}x$$
 [/INST] Substituting $\cos x = y$, the integral becomes
$$
\frac{1}{2} \int \frac{(-dy)}{2y^2-1} = \frac{1}{4} \int \frac{1}{\frac{1}{2}-y^2} dy.
$$
Since the integrand is a rational function of $y$, we can write it in terms of partial fractions:
$$
\frac{1}{\frac{1}{2}-y^2} = \frac{1}{\sqrt{2}} \left(\frac{1}{\frac{1}{\sqrt 2} +y} + \frac{1}{\frac{1}{\sqrt 2}-y} \right).
$$
Can you take it from here? 
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