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Integrals

Question
CBSEENMA12032658

Show that integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space straight g left parenthesis straight x right parenthesis space dx space equals space 2 space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx comma space space if space straight f and g are defined as f(x) = f(a - x) and g(x) + g(a-x) = 4

Solution

We have
f(x) = f(a-x)     ...(1)
and g (x) + g (a – x) = 4    ...(2)
Let I =                                                 ...(3)
      equals space integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis space straight g left parenthesis straight a minus straight x right parenthesis space 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
       equals space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space space open curly brackets 4 minus straight g left parenthesis straight x right parenthesis close curly brackets space dx                                           open square brackets because space of space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis close square brackets
       equals 4 integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space minus space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space straight g left parenthesis straight x right parenthesis space dx
therefore space space space space straight I space equals space 4 integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space minus space 1                                               left square bracket because space of space space left parenthesis 3 right parenthesis right square bracket
therefore space space 2 space straight I space equals space 4 integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx
therefore space space space space straight I space equals space 2 integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx

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