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Integrals

Question
CBSEENMA12032552

By using the properties of definite integrals, evaluate the following:
integral subscript 0 superscript straight pi over 2 end superscript space cos squared straight x space dx

Solution

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

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