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Math Help - Equivalent Integrals

  1. #1
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    Equivalent Integrals

    Given the definite integral \int_{\sqrt{2}}^{2} \frac{f(x)}{2\pi}dx, which of the following integrals are equivalent and justify your answer. (There could be more then one.)

    A. \int_{0}^{2\pi} f(\frac{x}{\pi})dx
    B. \int_{\frac{1}{4}}^{\frac{1}{2}}f(2sin(x\pi))cos(x  \pi)dx
    C. -\frac{1}{2\pi}\int_{1}^{2}\frac{f(\frac{1}{x})}{x^  2}dx

    I need some help on this one...
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  2. #2
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    Quote Originally Posted by ebonyscythe View Post
    Given the definite integral \int_{\sqrt{2}}^{2} \frac{f(x)}{2\pi}dx, which of the following integrals are equivalent and justify your answer. (There could be more then one.)

    A. \int_{0}^{2\pi} f(\frac{x}{\pi})dx
    B. \int_{\frac{1}{4}}^{\frac{1}{2}}f(2sin(x\pi))cos(x  \pi)dx
    C. -\frac{1}{2\pi}\int_{1}^{2}\frac{f(\frac{1}{x})}{x^  2}dx

    I need some help on this one...
    I'd suggest that you work backward: Take each answer and perform a substitution that will bring the argument of the function f back to x.

    For example, look at the first answer:
    \int_{0}^{2\pi} f(\frac{x}{\pi})dx

    Let u = \frac{x}{\pi} \implies du = \frac{dx}{\pi}

    So
    \int_{0}^{2\pi} f(\frac{x}{\pi})dx = \int_0^{2}f(u) \cdot \pi ~ du

    Change the "dummy" variable u to x and we get that
    \int_{0}^{2\pi} f(\frac{x}{\pi})dx = \pi \int_0^{2}f(x)~ dx
    which is not the same as the integral you are looking for.

    -Dan
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