Question

Integrate the following functions with respect to x :

`(1 + cos 4x)/(cos x - tan x)`

Answer 1

`int (1 + cos 4x)/(cos x - tan x) * "d"x = int (1 + cos 2(2x))/(cosx/sinx - sinx/cosx) * "d"x`

= `int (2cos^2 2x)/((sin^2x - cos^2x)/(sinx cos x)) * "d"x`

= `int (2sin x ocs x * cos^2 2x)/(cos^2x - sin^2x) * "d"x`

= `int (2sinx cosx * cos^2 2x)/(cos^2x - sin^2x) * "d"x`

= `int (sin 2x cos^2x)/(cos 2x) * "d"x`

= `int sin 2x cos 2x * "d"x`

= `int 1/2 xx 2 sin 2x cos 2x * "d"x`  .......[sin 2A = 2 sin A cos A]

= `1/2 int sin 4x * "d"x`

= `1/2 xx - (cos 4x)/4 + "c"`

= `- 1/8 cos 4x + "c"`