Integrate the following functions w.r.t. x.
∫1cos(x+a) cos(x+b)dx.
∫1cos(x+a) cos(x+b)dx
On multiplying and dividing by sin(a - b), we get
I=1sin(a−b)∫sin(a−b)cos(x+a) cos(x+b)dx⇒ I=1sin(a−b)∫sin[(x+a)−(x+b)]cos(x+a) cos(x+b)=1sin(a−b)∫sin(x+a)cos(x+b)−cos(x+a)sin(x+b)cos(x+a)cos(x+b)dx [∵ sin(A−B)=sin A cos B−sin B cos A]=1sin(a−b)∫[sin(x+a)cos(x+a)−sin(x+b)cos(x+b)]dx=1sin(a−b)∫[tan(x+a)−tan(x+b)]dx=1sin(a−b)[−log|cos(x+a)|+log|cos(x+b)|]+C=1sin(a−b)log ∣∣cos(x+b)cos(x+a)∣∣+C (∵ log m−log n=logmn)