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