Prove an identity of integral equations

Question

I tried to prove this: equation

And I came to: u=1-x

But now if I replace u by x-1, I get the same equation that needed to be proven. So I do not know actually what to do... Could someone help me?

Answer 1

The integral

$$\int_a^b f(x) dx$$

is the area under the curve of $f(x)$ as $x$ goes from $a$ to $b$.

The integral

$$\int_a^b f(u) du$$

is the area under the curve of $f(u)$ as $u$ goes from $a$ to $b$.

These are the same area: $x$ and $u$ are called dummy variables because it doesn't matter which letter you use, the area is the same, because you're adding up the same values of the same function in your Riemann sum. In a similar way,

$$\sum_{n=1}^\infty a_n = \sum_{k=1}^\infty a_k,$$

because again, $n$ and $k$ are simply dummy variables. One last example to make this notion clear: if you add up

$$\sum_{n=1}^3 n \qquad \textrm{or} \qquad \sum_{k=1}^3 k$$

you will get

$$1+2+3=6$$

from both sums. The takeaway: It does not matter what the dummy variable is called. It is irrelevant whether we already used a letter for a previous substitution.

Answer 2

You are done with the problem and you do not have to do anything else.

Note that x or u are dummy variables and you may simply change u to x as your last step.