Integration of (cosec^2 x-2005) : cos^2005 x dx is

We are asked in the problem to evaluate the integral of (cosec^2 x-2005) : cos^2005 x dx. The function is an example of a complex function with a degree that is greater than one and that uses special rules to integrate the function via the trigonometric functions. For example, we integrate

2005/cos^2005x dx which is equal to 2005 sec^2005 x since sec is the inverse of cos. The integral of this function when n >3 is equal to I = ∫ sec (n-2) xdx+∫tanx sec (n-3) x (secxtanx) dx

Then,

∫tanx sec (n-3) x (secxtanx) dx = tanx sec (n-2) x / (n-2) - 1 / (n-2) I

we can then integrate the function by substituting n by 3.

On the first term csc^2 2005x / cos^2005 x we can use the trigonometric identity csc^2 x = 1 + cot^2 x to simplify the terms