Let f(x)=2−sin⁡x2+cos⁡xStatement-1: The antiderivative of f is a periodic functionof period π.Statement-2: The period of the functiong(x)=log⁡(2+cos⁡x)+Atan−1⁡(Btan⁡x/2)+C,A,B,C being contants is 2π.

F(x)=2I1+I2,I1=∫dx2+cos⁡x and I2=−∫sin⁡x2+cos⁡xdx=log⁡(2+cos⁡x)+ constant. I1=∫2dt1+t22+1−t21+t2=2∫dt3+t2=23tan−1⁡13tan⁡(x/2)+C (t=tan⁡(x/2))Hence F(x)=log⁡(2+cos⁡x)+43tan−1⁡13tan⁡(x/2)+C which is periodic with period 2π.