Find the antiderivative of cos x + sin x?

-1/4 cos 2x + C 

well, sinxcosx=sin2x2 so you are looking at12∫ sin2x dx=(12)[(12)(−cos2x)+C]=−14cos2x+C'

or maybe easier you can notice the pattern that (sinnx)'=nsinn−1xcosx and pattern match. here n−1=1 so n = 2 so we trial (sin2x)' which gives us 2sinxcosx so we now that the anti deriv is 12sin2x+C

the other pattern also works ie(cosnx)'=ncosn−1x(−sinx)=−ncosn−1xsinx

so trial solution (−cos2x)'=−2cosx(−sinx)=2cosxsinx so the anti deriv is −12cos2x+C