Let f(x) be an indefinite integral of cos^3 x | vedclass

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(D) Statement-$2$: $\cos^3 x$ is a periodic function. This is a true statement because $\cos x$ is periodic with period $2\pi$,and any power of a peri

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Statement $1$ is false,Statement $2$ is true.

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(D) Statement-$2$: $\cos^3 x$ is a periodic function. This is a true statement because $\cos x$ is periodic with period $2\pi$,and any power of a periodic function is also periodic.Given $f(x) = \int \cos^3 x \, dx$.Using the identity $\cos 3x = 4\cos^3 x - 3\cos x$,we have $\cos^3 x = \frac{\cos 3x + 3\cos x}{4}$.$f(x) = \int \left(\frac{\cos 3x}{4} + \frac{3\cos x}{4}ight) dx = \frac{1}{4} \cdot \frac{\sin 3x}{3} + \frac{3}{4} \sin x + C = \frac{1}{12} \sin 3x + \frac{3}{4} \sin x + C$.The period of $\sin 3x$ is $\frac{2\pi}{3}$ and the period of $\sin x$ is $2\pi$.The period of the sum of two periodic functions is the least common multiple $(LCM)$ of their individual periods.Period of $f(x) = 	ext{LCM}\left(\frac{2\pi}{3}, 2\piight) = 2\pi$.Since the period is $2\pi$ and not $\pi$,Statement-$1$ is false.

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.

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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.