Question

Let $f\left(x\right)$ be an indefinite integral of ${\sin }^{2}x$.

Statement $I$: The function $F\left(x\right)$ satisfies $F\left(x+\mathrm{\pi }\right)=F\left(x\right)$ for all real $x$.

Statement II: ${\sin }^2\left(x+\mathrm{\pi }\right)={\sin }^{2}x$ for all real $x$.

• A. Statement $I$ is correct, Statement $II$ is correct; Statement $II$ is the correct explanation for Statement $I$.
• B. Statement $I$ is correct, Statement $II$ is correct; Statement $II$ is not a correct explanation for Statement $I$
• C. Statement $I$ is correct, Statement $II$ is correct
• D. Statement $I$ is incorrect, Statement $II$ is correct

Answer 1

The correct option is D

Statement $I$ is incorrect, Statement $II$ is correct

Explanation of correct answer :

Given, $F\left(x+\mathrm{\pi }\right)=F\left(x\right)$

$$\begin{gathered} F\left(x\right)=\int {\sin }^{2}xdx=\int \frac{1-\cos 2x}{2}dx \\ \Rightarrow F\left(x\right)=\frac{1}{4}\left(2x-\sin 2x\right)+c \\ \Rightarrow F(x+\mathrm{\pi })\ne F\left(x\right) \end{gathered}$$

Thus, Statement $I$ is false.

while, Statement $II$ is true as ${\sin }^{2}x$ is periodic with period $\mathrm{\pi }$.

Hence, the correct answer is Option(D).