Let f(x) be an indefinite integral of cos ^3 x. Statement 1: f(x) is a periodic function of period π. Statement 2: cos ^3 x is a periodic function.

Statement $-2: \cos ^3 x$−2:cos3x is a periodic function. it is a true statement. Statement-1 Given $\displaystyle f(x)=\int \cos ^3 x d x$f(x)=∫cos3xdx $$\begin{aligned} & =\int\left(\frac{\cos 3 x}{4}+\frac{3 \cos x}{4}\right) d x \\ & =\frac{1}{4} \frac{\sin 3 x}{3}+\frac{3}{4} \sin x \\ & =\frac{1}{12} \sin 3 x+\frac{3}{4} \sin x \end{aligned}$$​=∫(4cos3x​+43cosx​)dx=41​3sin3x​+43​sinx=121​sin3x+43​sinx​ Now, period of $\frac{1}{12} \sin 3 x=\frac{2 \pi}{3}$121​sin3x=32π​ Period of $\frac{3}{4} \sin x=2 \pi$43​sinx=2π Hence period of $f(x)=\frac{\text { L.C.M. }(2 \pi, 2 \pi}{\text { HCF of }(1,3)}$f(x)= HCF of (1,3) L.C.M. (2π,2π​ $$=\frac{2 \pi}{1}=2 \pi$$=12π​=2π Thus, $f(x)$f(x) is a periodic function of period $2 \pi$2π. Hence, Statement $-1$−1 is false.