Symmetric functions and definite integrals. Having trouble understanding the proof.

Question

I am reading this text about how to solve integrals for functions that are symmetric (even or odd):

Here's the proof:

Am I right that they left out some useful equivalences like: $dx = -du$? Because we eventually u substitute $dx$ for $-du$ right?

Also I am most confused about this line:

How did we get to: $$-\int_0^a f(-u)(-du)$$

But how did the -a change to an a? Is it because there are 2 ways to evaluate integrals, one of them being this concept:

How did the negatives cancel in the last part? Is it because the negative in front of $-du$ cancelled with the negative outside of the integral?

Also why are we subbing u for -x?

Also, it's weird that they write "we make the substitution $u = -x$ when really we're subbing the $x = -u$ right?

Answer 1

Hint:

The substitution $x=-u$ is the same as $u=-x$ and implies that $dx= -du$.

also if limit of integration is $x=a$ it becomes $-u=a$ that is equivalent to $u=-a$.

Answer 2

In the first, yes, they made the substitution u= -x and, I suppose, assumed that you could derive du= -dx from that yourself.

In the second, we have the integral $\int_0^{-a} f(-x)dx$. We make the substitution u= -x so du= -dx. f(-x)= f(-(-u))= f(u). When x= 0, u= 0 and when x= =a, u= -(-a). The integral changes to $\int_0^{a} f(u)(-du)= -\int_0^a f(u)du$

I would not think that it is "weird that they write "we make the substitution u=−x when really we're subbing the x=−u right?"

Surely, you understand that "u= -x" if and only if "x= -u". But this is really a question of grammar rather than mathematics. Saying "Substitute x= -u" [b]means[/b] "replace every -u with x". But that is not what we are doing because there was no "u" in the original integral. We are replacing every x with -u so we say "let u= -x".