Mathematics - Indefinite Integration Question with Solution | TestHub

Statement $-2:{\cos }^{3}x$ is a periodic function. it is a true statement. Statement-1 Given $f(x)=\int {\cos }^{3}xdx$ $$\begin{array}{rl}& =\int \left(\frac{\cos 3x}{4}+\frac{3\cos x}{4}\right)dx\\ & =\frac{1}{4}\frac{\sin 3x}{3}+\frac{3}{4}\sin x\\ & =\frac{1}{12}\sin 3x+\frac{3}{4}\sin x\end{array}$$ Now, period of $\frac{1}{12}\sin 3x=\frac{2\pi }{3}$ Period of $\frac{3}{4}\sin x=2\pi$ Hence period of $f(x)=\frac{\text{ L.C.M. }(2\pi ,2\pi }{\text{ HCF of }(1,3)}$ $$=\frac{2\pi }{1}=2\pi$$ Thus, $f(x)$ is a periodic function of period $2\pi$. Hence, Statement $-1$ is false.