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Need help with deciphering calculus 2 problem

  1. Feb 14, 2013 #1
    1. The problem statement, all variables and given/known data

    Use substitution to show that for any continuous function f,
    $$\int_0^{\pi/2} f(\sin x)\,dx = \int_0^{\pi/2} f(\cos x)\,dx.$$
    2. Relevant equations
    $$\cos(\pi/2-x)=\sin x$$

    3. The attempt at a solution

    My confusion is that f is inside the integral, and I have no idea if it would change anything.

    When I do the substitution normally, I get ##\int_0^{\pi/2} f(\cos u)\,du##, but that doesn't help because it is in terms of u and not x.

    Can someone help me understand this?
     
    Last edited by a moderator: Feb 14, 2013
  2. jcsd
  3. Feb 14, 2013 #2
    Shoot. I don't know how the formatting works on this site. The squigly line is an integrand and they all go from 0 to pi/2.
     
    Last edited: Feb 14, 2013
  4. Feb 14, 2013 #3

    eumyang

    User Avatar
    Homework Helper

    I don't think it matters. In the definite integrals of f(x) dx, from x = a to x = b, x is a dummy variable, since it could be replaced everywhere by any other letter and the meaning would be unchanged. So IIRC you can just change the letters from
    [tex]\int_0^{\pi/2} f(\cos u)\,du[/tex]
    to
    [tex]\int_0^{\pi/2} f(\cos x)\,dx[/tex]
    and you're done.
     
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