Question

If the value of the integral
$\int\limits_0^\pi {\frac{cos\ 4x-cos\ 4\alpha}{cos\ x-cos \alpha}} \, dx=K *\pi\ Cos 2\alpha\ Cos \alpha,$
Then the value of K is ____ ?

Answer 1

We can do this in a simple way as follows:

Let f(x) = (cos 4x - cos 4α) / (cos x - cos α)
  = (2cos^2 2x - 1 - 2 cos^2 2 α +1) / (cos x - cos α)
  = 2 (cos 2x + cos 2α) (cos 2x - cos2α) / (cos x - cos α)
  = 2(cos 2x+ cos 2α)  (2cos^2 x - 1 - 2cos^2 α +1) / (cos x -cos α)
  = 4 (cos 2x + cos 2α) (cos x + cos α)

As cos x, cos 2x have equal positive and negative values from 0 to Pi, the integral results in 0 for terms containing cos x or cos 2x.  The integral from x = 0 to pi/2 cancels the integral from x = pi/2 to pi.

Ans:  4 cos 2α cos α * [ x ]   for x = 0 to π

So K = 4.