Hey so uuuhh I need a lil help with question 7b)ii) from 2011 paper.
I got part i) easily, it was just a subsitution but then when I was looking at the solutions at the end of part i) they “relabelled” u as x so they could use it in their solution for part ii). Why is it that they can just relabel it as x straight after going through that whole process of substiting u=4-x?? Btw I tried to post screenshots of the question I was talking about but it said it was too large or something?
With the screenshots being too large, when that happens you'll need to upload it to an image hosting website like imgur and extract the link instead.
\[ u\text{ and }x\text{ are ultimately, nothing more than 'dummy variables'}\\ \text{used for the sake of definite integration.} \]
\[ \text{In general, it is always true that }\int_a^b f(x)\,dx = \int_a^b f(u)\,du.\\ \text{As an example, you can try computing }\int_0^1 x^2\,dx\\ \text{and check that it does equal }\int_0^1 u^2\,du.\]
Note that the fact they're equal is facilitated by the fact that \(x\) and \(u\) are 'doomed' to disappear, because once you sub the boundary values in, you don't care at all about what the variables you had at the start anymore were. This is the reason for why they're called 'dummy' variables in integration. Note that you don't run into the same luck when dealing with indefinite integrals, because the variables never disappear then.
\[ \text{So for that one, by the same logic,}\\ \int_1^3 \frac{\sin^2 \frac{\pi u}{8}}{u(4-u)}\,du = \int_1^3 \frac{\sin^2 \frac{\pi x}{8}}{x(4-x)}\,dx\]