∫cos4x dx = cos2x * cos2x dx = (1-sin2x)cos2x dx = cos2x dx - sin2x cos2x dx = 1/2( cos2x+1)dx - 1/4 (4sin2x cos2x dx) cos2x= 2cos2x - 1 cos2x = 1/2 (cos2x+1) = 1/2 cos2x dx + 1/2 1 dx - 1/4[sin22x dx] = 1/4 sin2x + x/2 - 1/8[1-cos4x dx] cos2x=1- 2sin2x sin^2 x=1/2(1-cos2x) = 1/4 sin2x + x/2 - 1/8 dx + 1/8 cos4x dx = 1/4 sin2x + x/2 - x/8 + 1/32 sin4x + C = 1/32 sin4x + 1/4 sin2x + 3x/8 + C

integration of cos⁴x

∫cos⁴x dx
= cos²x * cos²x dx
= (1-sin²x)cos²x dx
= cos²x dx - sin²x cos²x dx
= 1/2( cos2x+1)dx - 1/4 (4sin²x cos²x dx)                 cos2x= 2cos²x - 1   cos²x = 1/2 (cos2x+1)
= 1/2 cos2x dx + 1/2 1 dx - 1/4[sin²2x dx]
= 1/4 sin2x + x/2 - 1/8[1-cos4x dx]                           cos2x=1- 2sin²x       sin^{^2} x=1/2(1-cos2x)
= 1/4 sin2x + x/2 - 1/8 dx + 1/8 cos4x dx
=  1/4 sin2x + x/2 - x/8 + 1/32 sin4x + C
= 1/32 sin4x + 1/4 sin2x + 3x/8 + C

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integration of cos​4x

integration of cos⁴x