hii

let us first try to simplify the expression ...

 

so

(cos5x + cos4x)/(1 - 2cos3x)

= (2cos(9x/2).cos(x/2))/(1 - 2cos3x)
= (2cos(9x/2).cos(x/2))/( 1 - 2.(2cos²(3x/2) - 1) )

= (2cos(9x/2).cos(x/2))/(3 - 4.cos²(3x/2))

= (2.(4.cos³(3x/2) - 3cos(3x/2)).cos(x/2))/(3 - 4.cos²(3x/2))

= -2.cos(3x/2)cos(x/2)

= -( cos2x + cosx )

 

so now the integral is  -sin(2x)/2 - sinx + c

 

cheers