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Integrals

Question
CBSEENMA12036077

If g(x) = integral subscript 0 superscript straight x cos space 4 straight t space dt comma then g(x +π) equals

  • g(x)/g(π)

  • g(x) +g(π)

  • g (x) - g(π)

  • g(x). g(π)

Solution

B.

g(x) +g(π)

C.

g (x) - g(π)

Integral straight g space left parenthesis straight x right parenthesis space equals space integral subscript 0 superscript straight x space cos space 4 straight t. dt
To find g(x+π) in terms of g(x) of g(π)
straight g left parenthesis straight x right parenthesis space equals space integral subscript 0 superscript straight x space cos space 4 straight t space dt
rightwards double arrow space straight g left parenthesis straight x plus space straight pi right parenthesis space equals space integral subscript straight t space equals 0 end subscript superscript straight t space equals space straight x plus straight pi end superscript space cos space 4 straight t space dt
equals space integral subscript 0 superscript straight x space cos space 4 straight t space dt space plus space integral subscript straight x superscript straight x plus straight pi end superscript space cos space 4 straight t space dt
equals space straight g left parenthesis straight x right parenthesis space plus straight I subscript 1
straight l subscript 1 space equals space integral subscript straight x superscript straight x plus straight pi space end superscript space cos space 4 straight t space dt
equals integral subscript 0 superscript straight pi space cos space 4 straight t space dt
space equals space straight g left parenthesis straight pi right parenthesis
straight g left parenthesis straight x plus straight pi right parenthesis space equals space straight g left parenthesis straight x right parenthesis space plus straight g left parenthesis straight pi right parenthesis
But space the space value space of space straight I subscript 1 space is space zero
rightwards double arrow space straight g left parenthesis straight x plus straight pi right parenthesis space equals space straight g left parenthesis straight x right parenthesis minus straight g left parenthesis straight pi right parenthesis

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