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Integrals

Question
CBSEENMA12036063

Statement I - The value of the integral integral subscript straight pi divided by 6 end subscript superscript straight pi divided by 3 end superscript fraction numerator dx over denominator 1 plus space square root of tan space straight x end fraction is equal to π/6.
Statement II-integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript straight a superscript straight b straight f left parenthesis straight a plus straight b minus straight x right parenthesis space dx

  • Statement -I is True; Statement -II is True; Statement-II is a correct explanation for Statement-I

  • Statement - I is True; Statement -II is true; Statement-II is not a correct explanation for Statement-I

  • Statement -I is True; Statement -II is False.

  • Statement -I is False; Statement -II is True

Solution

D.

Statement -I is False; Statement -II is True

Let
straight I space equals space integral subscript straight pi divided by 6 end subscript superscript straight pi divided by 3 end superscript fraction numerator dx over denominator 1 plus square root of tan space straight x end root end fraction space.. space left parenthesis straight i right parenthesis
therefore space l space equals space space integral subscript straight pi divided by 6 end subscript superscript straight pi divided by 3 end superscript fraction numerator dx over denominator 1 plus square root of tan space open parentheses begin display style straight pi over 2 end style minus straight x close parentheses end root end fraction
rightwards double arrow l space equals space integral subscript straight pi divided by 6 end subscript superscript straight pi divided by 3 end superscript fraction numerator square root of tan space straight x end root space dx over denominator 1 plus square root of tan space straight x end root end fraction
On space adding space eq space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space get
2 l space equals space integral subscript straight pi divided by 6 end subscript superscript straight pi divided by 3 end superscript space dx
rightwards double arrow space 2 l space equals left square bracket straight x right square bracket subscript straight pi divided by 6 end subscript superscript straight pi divided by 3 end superscript
l italic space equals space 1 half open square brackets straight pi over 3 minus straight pi over 6 close square brackets space equals space straight pi over 12
Statement space straight I space is space false.
But space integral subscript straight a superscript straight b space straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript straight a superscript straight b space straight f left parenthesis straight a plus straight b minus straight x right parenthesis
a true statement by property definite integrals.

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