Question

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

• A. statemenstatement -1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
• B. Statement-1 is True, Statement-2 is True; Statement-2 is Not a correct explanation for statement-1
• C. statement - 1 is True, Statement - 2 is False
• D. Statement - 1 is False, Statement- 2 is True

Answer 1

Answer: (D)

Statement - 1 is False, Statement- 2 is True

$f\left(x\right)=\int {\sin }^{2}xdx=\frac{1}{2}\int (1-{\cos }^{2}x)dx=\frac{1}{2}(x-\frac{1}{2}\sin 2x)+c$
$\therefore f(x+\pi )\ne f\left(x\right)$
$\therefore$ Statement $1$ false
Statement$2:{\sin }^2(x+\pi )={\sin }^{2}x$ is true $\forall x\in R$