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Evaluate: int(cos5x+cos4x)/(1-2cos3x)dx...

Evaluate: `int(cos5x+cos4x)/(1-2cos3x)dx`

A

`(sin2x)/(2)+cosx+c`

B

`(sin2x)/(2)-cosx+c`

C

`-(sin2x)/(2)-sinx+c`

D

`(sin2x)/(2)-cosx+c`

Text Solution

Verified by Experts

The correct Answer is:
C
`I=int(cos5x+cos4x)/(1-2cos3x)dx=int(2"cos"(9x)/(2)"cos"(x)/(2))/(1-2(2"cos"^(2)(3x)/(2)-1))dx`
`=int(2"cos"(9x)/(2)"cos"(x)/(2))/(3-4"cos"^(2)(3x)/(2))dx`
`=int(2"cos"(9x)/(2)"cos"(x)/(2)"cos"(3x)/(2))/(3"cos"(3x)/(2)-4"cos"^(3)(3x)/(2))dx`
`=int(2"cos"(9x)/(2)"cos"(3x)/(2)"cos"(x)/(2))/(-"cos"(9x)/(2))dx`
`=-int 2"cos"(3x)/(2)"cos"(x)/(2)dx`
`=-int(cos2x+cosx)dx`
`=-(sin2x)/(2)-sinx+c`
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