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And here I got stuck. What should I do next?

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    Are you guys doing FIITJEE AITS? – 2017-02-26
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    It was given to us by our teacher. I don't know if it has been taken from AITS. – 2017-02-26
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    The nice thing of multiple choice tests is : If we can rule out all possibilities but one, we are done. Here, it is enough to bound the integral, you need not calculate it. Impressing that it is apparatly possible, but it is not necessary here (See answer and comment below) – 2017-02-26
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    Hey Rohan are/were you preparing for Jee ? – 2017-02-26

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Hint : $$|\int_{x=0}^\pi \frac{\cos(2017x)}{5-4\cos(x)}\mathrm {d}x|\le |\int_{x=0}^\pi\frac{1}{5-4\cos(x)}\mathrm {d}x|=\frac{\pi}{3}$$

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    To get the solution $D)$, there is an even easier way. The integrand of the second integral is positive and bounded by $1$ from above, so $\pi$ is an upper bound of the absolute value of the given integral. This is already enough to rule out $A)-C)$ – 2017-02-26
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    Which integral do you call the second integral? Is it the integral on the right in your answer. How did you say its bounded by 1 from above? – 2017-02-26
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    @Cotton I mean $\int_{x=0}^\pi \frac{1}{5-4\cos(x)}dx$. The integrand is positive and smaller than or equal to $1$ for every $x$ – 2017-02-26