int(cos4x-1)/(cotx-tanx)dx is equal to...

`int(cos4x-1)/(cotx-tanx)dx` is equal to

A

`(1)/(2)In|sec2x|-(1)/(4)cos^(2)2x+c`

B

`(1)/(2)In|sec2x|+(1)/(4)cos^(2)x+c`

C

`(1)/(2)In|cos2x|-(1)/(4)cos^(2)2x+c`

D

`(1)/(2)In|cos2x|+(1)/(4)cos^(2)x+c`

Text Solution

Verified by Experts

The correct Answer is:

C

`I=int(cos4x-1)/(cotx-tanx)dx=int(-2sin^(2)2x(sinxcosx))/((cos^(2)x-sin^(2)x))dx`
`=-int(sin^(2)2xsin2x)/(cos2x)x`
`=int((cos^(2)2x-1)sin2x)/(cos2x)dx`
`"Let " t=cos2x " or "dt=-2sin2xdx`
` :. I=(1)/(2)int((1-t^(2)))/(t)dt=(1)/(2)In|t|-(t^(2))/(4)+C `
`=(1)/(2)In|cos2x|-(1)/(4)cos^(2)2x+c`

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