Consider the given integral.
I=∫cos2xdx
Since,
cos2x=2cos2x−1
cos2x=cos2x+12
Therefore,
I=∫cos2x+12dx
I=12∫(cos2x+1)dx
I=12(sin2x2+x)+C
I=sin2x4+x2+C
Hence, this is the answer.