# Prove the following integral

1. Nov 17, 2014

### FlorenceC

1. The problem statement, all variables and given/known data
prove that the ∫01 f(x) = ∫01f(1-x)

2. Relevant equations

3. The attempt at a solution
I got all the way to ∫01-f(u) du where u = 1-x but I don;t know how to prove it.

2. Nov 17, 2014

### ehild

When you substitute the integration variable the limits also change.

3. Nov 17, 2014

### FlorenceC

Okay, so -∫1 f(u) du = ∫01f(u) du. but how do i relate this to ∫f(x)?

4. Nov 17, 2014

### ehild

It does not matter how do you name the integration variable. It can be x instead of u. The function is the same, and so are the limits.

5. Nov 17, 2014

### Ray Vickson

Don't you see that $\int_0^1 f(x) \, dx = \int_0^1 f(u) \, du = \int_0^1 f(\text{anything}) \, d\text{anything}$?

6. Nov 17, 2014

### Staff: Mentor

To elaborate on what Ray said, I have added variables in the limits of integration to emphasize that in each integral we have a different dummy variable.
$$\int_{x = 0}^1 f(x) \, dx = \int_{u = 0}^1 f(u) \, du = \int_{\text{anything} = 0}^1 f(\text{anything}) \, d\text{anything}$$

When you use substution to change the variable of integration, you need to either change the limits of integration (which was not done in the above) or undo the substitution.