`"If " int(cos 4x+1)/(cotx-tanx)dx=Acos4x+B, " then "`

Options

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`

Answer

C

Explanation

`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`