Question

3. cos 2x cos 4x cos 6x

Answer 1

The given function is cos2xcos4xcos6x

∫ cos2xcos4xcos6xdx (1)

Also, cosAcosB= 1 2 { cos(A+B)+cos( A−B ) }(2)

Equation (1) can be written as,

∫ cos2x(cos4xcos6x)dx (3)

Use (2) to further solve the equations,

∫ cos2x(cos4xcos6x)dx = ∫ cos2x [ 1 2 { cos(4x+6x)+cos( 4x−6x ) } ]dx = 1 2 ∫ { cos2xcos10x+cos2xcos(−2x) }dx = 1 2 ∫ { cos2xcos10x+ cos 2 2x }dx = 1 2 ∫ [ { 1 2 cos( 2x+10x )+cos( 2x−10x ) }+( 1+cos4x 2 ) ] dx

Further simplify the equations to integrate the given function,

∫ cos2x(cos4xcos6x)dx = 1 4 ∫ ( cos12x+cos8x+1+cos4x )dx = 1 4 [ sin12x 12 + sin8x 8 + sin4x 4 ]

Thus, the integral of the function cos2xcos4xcos6x is 1 4 [ sin12x 12 + sin8x 8 + sin4x 4 ].