Evaluate the following integrals : ∫1/(cos 3x - cos x) dx ​

Question

Evaluate the following integrals :

\(\int\frac{1}{cos\,3x-cos\,x}\)dx

Answer 1

The denominator is of the form cosC - cosD 

= - 2sin(\(\frac{c+d}{2}\)).sin (\(\frac{c-d}{2}\))

∴ cos3x - cosx = - 2sin(\(\frac{3+1}{2}x\)).sin (\(\frac{3-1}{2}x\))

∴ cos3x - cosx = - 2sin2x.sinx 

- 2sin2x.sinx = - 2.2.sinx.cosx.sinx 

- 2sin2x.sinx= - 4sin²x.cosx 

Also, 

sin²x + cos²x = 1