Let f(x) be an indefinite integral of ^3x.

Question

Let $f(x)$ be an indefinite integral of $\cos^3x$.

Statement 1: $f(x)$ is a periodic function of period $\pi$.

Statement 2: $\cos^3x$ is a periodic function.

Accepted Answer

Statement-2: $\cos^3x$ is a periodic function --- true.

$f(x) = \int \cos^3x\,dx = \int \left(\dfrac{\cos3x}{4}+\dfrac{3\cos x}{4}\right)dx = \dfrac{1}{12}\sin3x+\dfrac34\sin x$

Period of $\dfrac{1}{12}\sin3x$ is $\dfrac{2\pi}{3}$; period of $\dfrac34\sin x$ is $2\pi$

Period of $f(x)$ = LCM of $\left(\dfrac{2\pi}{3},2\pi\right) = 2\pi$

Thus $f(x)$ is periodic with period $2\pi$, not $\pi$. Statement-1 is false.

Solutions · 1