Find the antiderivative of cos x + sin x?

-1/4 cos 2x + C

or 1/2 sin^2 x + C

or - 1/2 cos^2 x + C

well, sinx cosx = sin2 x 2 so you are looking at 12 ∫ sin2 x dx = (12) [ (12) ( - cos2 x) + C] = - 14 cos2 x+C'

or maybe easier you can notice the pattern that (sinn x) '=n sin n-1 x cosx and pattern match. here n-1=1 so n = 2 so we trial (sin2 x) ' which gives us 2 sinx cosx so we now that the anti deriv is 12 sin2 x + C

the other pattern also works ie (cosn x) '=n cos n-1 x ( - sinx) = - n cos n-1 x sinx

so trial solution ( - cos2 x) '=-2 cosx ( - sinx) = 2 cosx sinx so the anti deriv is - 12 cos2 x + C