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Question

3. cos 2x cos 4x cos 6x

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Solution

The given function is cos2xcos4xcos6x

cos2xcos4xcos6xdx (1)

Also, cosAcosB= 1 2 { cos(A+B)+cos( AB ) }(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( 4x6x ) } ]dx = 1 2 { cos2xcos10x+cos2xcos(2x) }dx = 1 2 { cos2xcos10x+ cos 2 2x }dx = 1 2 [ { 1 2 cos( 2x+10x )+cos( 2x10x ) }+( 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 ].


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