If $integral subscript 0 superscript straight pi space xf space left parenthesis sin right parenthesis space dx space equals space straight A space integral subscript 0 superscript straight pi divided by 2 end superscript space straight f space left parenthesis sin space straight x right parenthesis space dx comma space then space straight A space is$

0

π

π/2

π/4

Solution

Let $let space straight I space equals space integral subscript 0 superscript straight pi space xf space left parenthesis sin space straight x right parenthesis space dx space equals space integral subscript 0 superscript straight pi space left parenthesis straight pi minus straight x right parenthesis space straight f space left parenthesis sin space straight x right parenthesis space dx space equals space straight pi integral subscript 0 superscript straight pi space straight f space left parenthesis sin space straight x right parenthesis dx minus 1 left parenthesis since space straight f space left parenthesis 2 straight a minus straight x right parenthesis space equals space straight f left parenthesis straight x right parenthesis right parenthesis rightwards double arrow space straight I space equals space straight pi integral subscript 0 superscript straight pi divided by 2 end superscript space straight f space left parenthesis sin right parenthesis space dx rightwards double arrow space straight I subscript 2 divided by straight I subscript 1 space equals space 2$
=π

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