Question

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.

• A. Statement 1 is true, Statement 2 is false
• B. Both the Statements are true, but Statement 2 is not the correct explanation of Statement 1
• C. Both the Statements are true, and Statement 2 is correct explanation of Statement 1
• D. Statement 1 is false, Statement 2 is true

Answer 1

Answer: (D)

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