Question

Evaluate $integral subscript 0 superscript straight pi divided by 2 end superscript space sin squared straight x space dx$

Solution

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

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Integrals

Evaluate $integral subscript 0 superscript straight pi divided by 2 end superscript space sin squared straight x space dx$

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