Substitution rule for integrals

Mar 2017
356
3
Massachusetts
Hi,

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

The question is as follows: Screen Shot 2019-07-21 at 10.43.26 AM.png

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:

IMG_20190721_104027.jpg

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
Nov 2013
6,666
3,004
California
$I=\displaystyle \int_{-a}^0 f(x)~dx = \int_{-a}^0 -f(-x)~dx$

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.
 
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