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

Arishta
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  • Are you guys doing FIITJEE AITS? –  Feb 26 '17 at 10:56
  • It was given to us by our teacher. I don't know if it has been taken from AITS. – Arishta Feb 26 '17 at 10:57
  • 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) – Peter Feb 26 '17 at 11:10
  • Hey Rohan are/were you preparing for Jee ? – Archis Welankar Feb 26 '17 at 12:57

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

Peter
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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 $A)-C)$ – Peter Feb 26 '17 at 11:03
  • 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? – Arishta Feb 26 '17 at 13:14
  • @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 $x$ – Peter Feb 26 '17 at 13:19