Find Integration of this trigonometric expression (Cos7x—Cos8x)/(1+2Sin5x)

cos 7x – cos 8x
= ( cos 7x + cos 3x ) – ( cos 8x + cos 2x ) + ( cos 2x – cos 3x )
= ( 2 cos 5x cos 2x ) – ( 2 cos 5x cos 3x ) + ( cos 2x – cos 3x )
= ( 2 cos 5x )( cos 2x – cos 3x ) + ( cos 2x – cos 3x )
= ( cos 2x – cos 3x ) ( 2 cos 5x + 1 )

divide it by 1+2cos5x to get integrand= cos2x – cos3x. now integrating we get

(1/2)sin2x – (1/3)sin3x + C.