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Integrals

Question
CBSEENMA12032582

Show that
integral subscript 0 superscript straight pi over 2 end superscript space sin squared straight x space dx space equals space straight pi over 4

Solution

Let I =  integral subscript 0 superscript straight pi divided by 2 end superscript sin squared straight x space dx                                             ...(1)
therefore space space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript sin squared open parentheses straight pi over 2 minus straight x close parentheses dx                     open square brackets because space space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis space dx close square brackets
therefore space space space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript cos squared straight x space dx                                         ...(2)
Adding (1) and (2). we get.
          2 space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript left parenthesis sin squared straight x space plus cos squared straight x right parenthesis space dx space equals space integral subscript 0 superscript straight pi divided by 2 end superscript 1 space dx space equals space open square brackets straight x close square brackets subscript 0 superscript straight pi divided by 2 end superscript space equals space straight pi over 2 minus 0 space equals space straight pi over 2
therefore space space space 2 space space straight I space equals space straight pi over 2 space space space space space space space rightwards double arrow space space space space straight I space equals space straight pi over 4

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