Let $F(x)$ be an indefinite integral of $\sin^2 x$. \ STATEMENT 1 : The function $F(x)$ satisfies $F(x+\pi) = F(x)$ for all real x. because \ STATEMENT 2 : $\sin^2(x+\pi) = \sin^2 x$ for all real x.

Let F(x) be an indefinite integral of sin² x. STATEMENT 1 : The function F(x) satisfies F(x+π) = F(x) for all real x. because STATEMENT 2 : sin²(x+π) = sin² x for all real x. $F(x) = \displaystyle\int \sin^{2}x\,dx$ $\text{Statement 1: } F(x+\pi) = F(x) \ \forall x \in \mathbb{R}$ $\text{Statement 2: } \sin^{2}(x+\pi) = \sin^{2}x \ \forall x \in \mathbb{R}$ $\text{To find: truth of both statements, and whether (2) explains (1)}$ $1.\ \text{find } F(x) = \displaystyle\int \sin^{2}x\,dx$ $2.\ \text{check Statement 2: } \sin^{2}(x+\pi)$ $3.\ \text{check Statement 1: compare } F(x+\pi) \text{ with } F(x)$ $F'(x) = \sin^{2}x, \qquad F(x) = \displaystyle\int \sin^{2}x\,dx$ $\sin^{2}x = \dfrac{1 - \cos 2x}{2}$ $F(x) = \dfrac{1}{2}\displaystyle\int(1-\cos 2x)\,dx = \dfrac{x}{2} - \dfrac{\sin 2x}{4} + C$ $\sin(x+\pi) = -\sin x \Rightarrow \sin^{2}(x+\pi) = \sin^{2}x \Rightarrow \text{Statement 2 is TRUE}$ $F(x+\pi) = \dfrac{x+\pi}{2} - \dfrac{\sin(2x+2\pi)}{4} + C$ $= \dfrac{x}{2} + \dfrac{\pi}{2} - \dfrac{\sin 2x}{4} + C = F(x) + \dfrac{\pi}{2}$ $F(x+\pi) - F(x) = \dfrac{\pi}{2} \neq 0 \Rightarrow \text{Statement 1 is FALSE}$ $\text{Mean of } \sin^{2}x \text{ over a period} = \tfrac12 \neq 0 \Rightarrow \text{a linear term } \tfrac{x}{2} \text{ survives}$ $\text{Statement 1 False, Statement 2 True} \Rightarrow (4)$ $\boxed{\text{Answer: (4)}}$