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Integrals

Question
CBSEENMA12032556

Evaluate the following:
integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 1 plus tan cubed straight x end fraction.

Solution

Let I = integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 1 plus tan cubed straight x end fraction                                  ...(1)
therefore space space space straight I space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 1 plus tan cubed open parentheses begin display style straight pi over 2 end style minus straight x close parentheses end fraction                 open square brackets because space space space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space space equals integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis space dx close square brackets
          equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 1 plus cot cubed straight x end fraction space equals space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator 1 plus begin display style 1 over tan cubed end style straight x end fraction
                      ...(2)
Adding (1) and (2), we get.
             
                   equals space integral subscript 0 superscript straight pi over 2 end superscript space 1 space dx space equals space open square brackets straight x close square brackets subscript 0 superscript straight pi over 2 end superscript space equals space straight pi over 2 minus 0 space equals straight pi over 2
   space therefore space space space 2 space straight I space equals space straight pi over 2 comma space space space space straight I space equals space straight pi over 4
therefore space space space integral subscript 0 superscript straight pi over 2 end superscript space fraction numerator dx over denominator 1 plus tan cubed straight x end fraction space equals space straight pi over 4

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