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Integrals

Question
CBSEENMA12036037
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The integral integral subscript 2 superscript 4 fraction numerator log space straight x squared over denominator log space straight x squared space plus space log space left parenthesis 36 minus 12 space straight x plus straight x squared right parenthesis end fraction dx  is equal to 

  • 2

  • 4

  • 1

  • 6

Solution

C.

1

straight I space equals space integral subscript 2 superscript 4 fraction numerator log space straight x squared over denominator log space straight x squared space plus space log space left parenthesis 36 minus 12 space straight x plus straight x squared right parenthesis end fraction dx space equals space integral subscript 2 superscript 4 fraction numerator 2 space log space straight x over denominator space 2 space log space straight x space plus space log space left parenthesis 6 minus straight x squared right parenthesis end fraction dx space equals space integral subscript 2 superscript 4 fraction numerator 2 space log space straight x space dx over denominator 2 left square bracket log space straight x space plus space log space left parenthesis 6 minus straight x right parenthesis right square bracket end fraction dx space space space.. space left parenthesis straight i right parenthesis space equals space integral subscript 2 superscript 4 fraction numerator log space left parenthesis 6 minus straight x right parenthesis over denominator log space left parenthesis 6 minus straight x right parenthesis space plus space log space straight x end fraction dx space space space... space left parenthesis ii right parenthesis open square brackets because space 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 space left parenthesis straight a plus straight b minus straight x right parenthesis dx close square brackets On space adding space eqs. space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space get 2 straight I space equals space integral subscript 2 superscript 4 fraction numerator log space straight x space plus space log space left parenthesis 6 minus straight x right parenthesis over denominator log space straight x space plus space log space left parenthesis 6 minus straight x right parenthesis end fraction dx rightwards double arrow space 2 straight I space equals space integral subscript 2 superscript 4 space dx space equals space left square bracket straight x right square bracket subscript 2 superscript 4 2 straight I space equals space 2 straight I space equals space 1

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