Question

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

Answer 1

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+∫tanxsec(n−3)x(secxtanx)dx
Then, 
∫tanxsec(n−3)x(secxtanx)dx=tanxsec(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