Typical Indefinite Integration!

Calculus Level 5

cos(2016x)sin(x)dx \large \int \frac{\cos(2016x)}{\sin(x)} \, dx

Let xx be a real number except at multiples of π\pi. Ignoring the arbitrary constant, if the indefinite integral is equals to

I=α(n=11008cos((2n1)x)2n1)+lntan(2xβ) \large I = \alpha \left( \sum_{n=1}^{1008} \frac{ \cos((2n-1)x)}{2n-1} \right) + \ln \left| \tan \left( \frac {2x}\beta \right) \right |

For constants α\alpha and β\beta , evaluate αββα\alpha^\beta - \beta^\alpha .

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