Let F(x) be an indefinite integral of ^2 x. STATEMENT -1 : The fu… - Vidyadip

MCQ

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 $x$.

A
Statement -$1$ 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
✓
Statement -$1$ is False, Statement -$2$ is True

✓

Answer

Correct option: D.

Statement -$1$ is False, Statement -$2$ is True

d
$ F(x)=\int \sin ^2 x d x=\int \frac{1-\cos 2 x}{2} d x $

$ \Rightarrow F(x)=\frac{1}{4}(2 x-\sin 2 x)+c$

Since, $F(x+\pi) \neq F(x)$.

Hence statement $1$ is false.

But statement $2$ is true as $\sin ^2 x$ is periodic with period $\pi$.

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