The value of the `int(sinx+cosx)/(3+sin2x)dx` is equal to

Options

A. `(1)/(4)log((2-sin x+cosx)/(2+sinx-cosx))+C`
B. `(1)/(2)log((2+sinx)/(2-sinx))+C`
C. `(1)/(4)log((1sinx)/(1-sinx))+C`
D. None of these

Answer

A

Explanation

Let `l=int(sinx+cosx)/(3+sin2x)dx=int(sinx+cosx)/(4+sin2x-1)dx`
`rArr" "l=int(sinx+cosx)/(4-(sinx-cosx)^(2))dx`
`"Put "sinx-cosx=t`
`rArr (cos x+sinx)dx=dt`
`therefore" "l=int(dt)/(4-t^(2))=(1)/(4)log((2-t)/(2+t))+C`
`" "=(1)/(4)log((2-sin x+cosx)/(2+sinx-cosx))+C`