I have only with one word a problem. In step three you say "obtain". If I obtain I got everything. "Obtain" is the problem.

The rule of substitution in general says:

Is $\displaystyle \varphi : [a,b] \rightarrow \mathbb{R}$ continous differentiable and $\displaystyle I \subset \mathbb{R}$ an intervall with $\displaystyle \varphi([a,b]) \subset I $ and $\displaystyle f: I \rightarrow \mathbb{R}$ piecewise continous, than is

$\displaystyle \int\limits_{\varphi(a)} ^{\varphi(b)} f(x) \mathrm{d}x = \int\limits_{a}^{b} f(\varphi(t)) \cdot \varphi \prime (t) \mathrm{d}t $

I understand the scentence and the proof of it but I can not apply it to the example above. The other $\displaystyle x$ is not an argument of $\displaystyle u$, so why we can obtain this? We know nothing about the area of definition here.

Argh, I guess I am too stupid to see the answer