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Integrals

Question
CBSEENMA12032660
Wired Faculty App

Evaluate: 
integral subscript 0 superscript 1 cot to the power of negative 1 end exponent left parenthesis 1 minus straight x plus straight x squared right parenthesis dx.

Solution

Let
I = integral subscript 0 superscript 1 cot to the power of negative 1 end exponent left parenthesis 1 minus straight x plus straight x squared right parenthesis dx
  equals space integral subscript 0 superscript 1 tan to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator 1 minus straight x plus straight x squared end fraction close parentheses dx space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space cot to the power of negative 1 end exponent straight x space equals space tan to the power of negative 1 end exponent 1 over straight x close square brackets
   equals space integral subscript 0 superscript 1 tan to the power of negative 1 end exponent open square brackets fraction numerator straight x plus left parenthesis 1 minus straight x right parenthesis over denominator 1 minus straight x left parenthesis 1 minus straight x right parenthesis end fraction close square brackets dx space equals space integral subscript 0 superscript 1 open square brackets tan to the power of negative 1 end exponent straight x plus tan to the power of negative 1 end exponent left parenthesis 1 minus straight x right parenthesis close square brackets dx space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space tan to the power of negative 1 end exponent straight x plus tan to the power of negative 1 end exponent straight y space equals space tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus straight y over denominator 1 minus xy end fraction close parentheses close square brackets equals space integral subscript 0 superscript 1 space tan to the power of negative 1 end exponent xdx space plus space integral subscript 0 superscript 1 tan to the power of negative 1 end exponent left parenthesis 1 minus straight x right parenthesis space dx equals space integral subscript 0 superscript 1 tan to the power of negative 1 end exponent straight x space dx plus integral subscript 0 superscript 1 tan to the power of negative 1 end exponent left curly bracket 1 minus left parenthesis 1 minus straight x right parenthesis right curly bracket space dx equals space integral subscript 0 superscript 1 tan to the power of negative 1 end exponent straight x space dx plus integral subscript 0 superscript 1 tan to the power of negative 1 end exponent straight x space dx space equals space 2 integral subscript 0 superscript 1 tan to the power of negative 1 end exponent straight x. space 1 space dx
    equals space 2 open square brackets left parenthesis tan to the power of negative 1 end exponent space straight x right parenthesis space left parenthesis straight x right parenthesis close square brackets subscript 0 superscript 1 space minus space 2 integral subscript 0 superscript 1 fraction numerator 1 over denominator 1 plus straight x squared end fraction cross times straight x space dx equals 2 space open square brackets straight x space tan to the power of negative 1 end exponent straight x close square brackets subscript 0 superscript 1 space minus space integral subscript 0 superscript 1 fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction dx space equals space 2 open square brackets straight x space tan to the power of negative 1 end exponent straight x close square brackets subscript 0 superscript 1 space minus space open square brackets log left parenthesis 1 plus straight x right parenthesis squared close square brackets subscript 0 superscript 1 equals space 2 left square bracket tan to the power of negative 1 end exponent space 1 space minus space 0 right square bracket space minus space left square bracket log space 2 space minus space log space 1 right square bracket equals space 2 open parentheses straight pi over 4 minus 0 close parentheses space space minus space left parenthesis log space 2 space minus space 0 right parenthesis space equals space straight pi over 2 minus log space 2

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