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Need help with deciphering calculus 2 problem

  • #1

Homework Statement



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

Homework Equations


$$\cos(\pi/2-x)=\sin x$$

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?
 
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Answers and Replies

  • #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:
  • #3
eumyang
Homework Helper
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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.
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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