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Let F(x) be an indefinite integral of si...

Let F(x) be an indefinite integral of `sin^(2)x`
Statement I The function F(x) satisfies `F(x+pi)=F(x)` for all real x.
Because
Statement II `sin^(2)(x+pi)=sin^(2)x,` for all real x.

A

Statement - 1 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 a correct explanation for Statement -1.

C

Statement - 1 True ,Statement - 2 is False.

D

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

Text Solution

Verified by Experts

It is given that `F(x)=intsin^(2)x dx=(1)/(2)int(1-cos2x)dx=(1)/(2)(x-(sin2x)/(2))+C`
`thereforeF(pi+x)-F(x)=(1)/(2)[{(pi+x)-(1)/(2)sin2(pi+x)}-{x-(1)/(2)sin2x}]`
`=(1)/(2)(pi-(1)/(2)sin2x+(1)/(2)sin2x)=(pi)/(2)` for all x thus,F `(x+pi)` = F (x) for all x is not true.
So , statement -2 is not true.
Clearly , statement -2 is true as `sin^(2)` a is periodic with period `pi`.
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Knowledge Check

  • Let f(x) be an indefinite integral of sin^(2)x . Consider the following statements : Statements 1. The function f(x) satisfies f(x+pi)=f(x) for all real x. 2. Sin^(2)(x+pi)=sin^(2)x for all real x. Which one of the following is correct in respect of the above statements ?

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