Integral test for convergence.The problem point in the denominator

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$$\int_{0}^{\frac{\pi}{2}} \frac{2 \cot{\left (x \right )} + 1}{2 \sin{\left (x \right )} + \cos{\left (x \right )}}\, dx$$

How to solve this problem? What are the main steps in solving this ? I understand that this integral should be wrtitten as a summ of integrals but I can't find the problem point.Is it this :

                           2*sin(x)+cos(x)=0

                           cos(x)/2sin(x)=-1

                             ctg(x)=-2
                             x=arcctg(-2)