Let $f(x)$ be an indefinite integral of $\cos ^3 x$. Statement 1: $f(x)$ is a periodic function of period…
Let $f(x)$ be an indefinite integral of $\cos ^3 x$.
Statement 1: $f(x)$ is a periodic function of period
$\pi$.
Statement 2: $\cos ^3 x$ is a periodic function.
Statement 1 is true, Statement 2 is false.
Both the Statements are true, but Statement 2 is not the correct explanation of Statement 1.
Both the Statements are true, and Statement 2 is correct explanation of Statement 1.
Statement 1 is false, Statement 2 is true.
Solution
Statement $-2: \cos ^3 x$ is a periodic function. it is a true statement.
Statement-1
Given $f(x)=\int \cos ^3 x d x$
$
\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}
$
Now, period of $\frac{1}{12} \sin 3 x=\frac{2 \pi}{3}$
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