Evaluate $\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx $

Question

Evaluate the indefinite integral
$$\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx$$

I tried multiplying and dividing by $\sec^2 {x} $ and then setting $\tan{x}=y$ but no good. Then I set $\cos {x}=t $ and tried to create $\sin {x} $ in the numerator. But the integral which came was also a difficult one.

Please Help!

Thanks!

P.S. I am a high school student so kindly use elementary methods only. Thanks again!

Answer 1

HINT:

Integrate by parts $$\int(\sec^{2005}x\cdot\csc^2x)dx$$

$$=\sec^{2005}x\int\csc^2x\ dx-\int\left(\frac{d(\sec^{2005}x)}{dx}\int\csc^2x\ dx\right)dx$$