Question

is integral f(-x)(-dx) = integral f(x)dx ??
let's assume the graph is continuous on real numbers and f(x) >= 0

Answer 1

Let's see:
\[\int f(-x)~(-dx)=-\int f(-x)~dx\overbrace{=}^{?}\int f(x)~dx\]
You basically want to show that \(f(-x)=-f(x)\). For what class of functions is this true?

Answer 2

I was trying to do the question 87, and I got stuck on the solution. it says integral f(u)du = integral f(x)dx, on the solution.
but if I substitute f(u)du with u = -x and du = -dx, f(-x)(-dx) comes out, which should equal to f(x)dx.

Answer 3

basically I didn't understand how f(u)du = f(x)dx because u = -x

Answer 4

Well that's not the same question that you posted.

Given \(\displaystyle\int_a^b f(-x)~dx\), substituting \(u=-x\) gives you \(\displaystyle-\int_{-a}^{-b}f(u)~du=\int_{-b}^{-a}f(u)~du\).

I would say that the book is then thinking of \(u\) as a dummy variable, meaning that you can use any variable to represent it.

Answer 5

In other words, forget the substitution you made; this new \(x\) is not the same as the old \(x\). Confusing, I know, but this explanation is very hand-wavy.