Need help with deciphering calculus 2 problem

[Al3x L3g3nd]

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?

 

[Al3x L3g3nd]

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.

 

[eumyang]

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
$$\int_0^{\pi/2} f(\cos u)\,du$$
to
$$\int_0^{\pi/2} f(\cos x)\,dx$$
and you're done.