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Integrals

Question
CBSEENMA12036080

The value of integral subscript 0 superscript 1 fraction numerator 18 space log space left parenthesis 1 plus straight x right parenthesis over denominator 1 plus straight x squared end fraction space dx is

  • straight pi over 8 space log space 2
  • straight pi over 2 space log space 2
  • log 2

  • π log 2

Solution

D.

π log 2

straight I space equals integral subscript 0 superscript 1 fraction numerator 18 space log space left parenthesis 1 plus straight x right parenthesis over denominator 1 plus straight x squared end fraction d x 
put x = tan θ
⇒ dx = sec2 θdθ
When x = 0 
⇒ tan θ = 0 
∴ θ = 0
When x = 1 = tan θ
θ = π/4
straight I space equals integral subscript 0 superscript 1 fraction numerator 18 space log space left parenthesis 1 plus tan space straight theta right parenthesis over denominator 1 plus tan squared space straight theta end fraction sec squared straight theta space dθ
straight I space equals space 8 integral subscript 0 superscript straight pi divided by 4 end superscript log space left parenthesis 1 plus space tan space straight theta right parenthesis space dθ space.... space left parenthesis straight i right parenthesis space
Using space space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 0 superscript straight a space straight f left parenthesis straight a minus straight x right parenthesis space dx comma space we space get
straight I space equals space 8 space integral subscript 0 superscript straight pi divided by 4 end superscript space log space open curly brackets 1 plus space tan space open parentheses straight pi over 4 minus straight theta close parentheses close curly brackets dθ
space equals space straight I space equals space 8 space integral subscript 0 superscript straight pi divided by 4 end superscript space log space open curly brackets 1 plus space fraction numerator 1 minus space tan space straight theta over denominator 1 plus space tan space straight theta end fraction close curly brackets dθ
equals space 8 space integral subscript 0 superscript straight pi divided by 4 end superscript space log space open curly brackets 1 plus space fraction numerator 2 over denominator 1 plus space tan space straight theta end fraction close curly brackets dθ space... space left parenthesis ii right parenthesis
Adding space eqs space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis thin space we space get
2 straight I space space equals space 8 space space integral subscript 0 superscript straight pi divided by 4 end superscript space log space 2 space dθ space equals space 4. space log space left parenthesis straight theta right parenthesis subscript 0 superscript straight pi divided by 4 end superscript
equals space 4 space log space 2. space open parentheses straight pi over 4 minus 0 close parentheses space equals space straight pi space log space 2 

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