# Substitution rule for integrals

#### otownsend

Hi,

I hope I can get some help with solving a problem.

The question is as follows:

Intuitively, I know the answer is 0.

Now I am in the process of proving it and I'm stuck.

Here is the work that I have done so far:

What I am confused about: do you interpret the problem as though f has an inner function, and that inner function being x? And if so, it's still not clear to me why you are "allowed" to assign u (the substitution variable) to - x. I see how this assignment is helpful towards solving the problem, but I don't see how this is allowed to be done in the first place, given my current understanding of the substitution rule. I see how assigning u to x would be valid, but not -x.

Help would be appreciated!

#### romsek

MHF Helper
$I=\displaystyle \int_{-a}^0 f(x)~dx = \int_{-a}^0 -f(-x)~dx$

Now the $-x$ appears explicitly in the integral. This may make it easier for you to accept

$u=-x,~du=-dx$

$I= \displaystyle \int_a^0 f(u)du = -\int_0^a f(u)~du = -\int_0^a f(x)~dx$

The last equality is because $u$ and $x$ are just dummy variables in the integral so we can call them whatever we like.

1 person

#### otownsend

Thank you so much!