cos^x^6

= (cosx²)³

= [(1+cos2x)/2]³ = (1/8)(1+cos2x)³

= (1/8)(1+3cos2x+3cos²2x+cos³2x)

= (1/8)+(3/8)cos2x+(3/8)(1/2)(1+cos4x)+(1/8)(1/2)(1+cos4x)(cos2x)

= (1/8)+(3/8)cos2x+(3/16)+(3/16)cos4x+(1/16)(cos2x+cos4xcos2x)

= (5/16)+(3/8)cos2x+(3/16)cos4x+(1/16)cos2x+(1/32)cos6x+(1/32)cos2x

= (5/16)+(15/32)cos2x+(3/16)cos4x+(1/32)cos6x

∫cos^6x dx = ∫[(5/16)+(15/32)cos2x+(3/16)cos4x+(1/32)cos6x] dx

= (5/16)∫ dx + (15/32)∫cos2x dx + (3/16)∫cos4x dx + (1/32)∫cos6x dx

= (5/16)∫ dx + (15/32)(1/2)∫cos2x d(2x) + (3/16)(1/4)∫cos4x d(4x) + (1/32)(1/6)∫cos6x d(6x)

= (5/16)x + (15/64)sin2x + (3/64)sin4x + (1/192)sin6x + C