cyt91 said:
What is invariance w.r.t. integral function anyway?
I'm not familiar with the phrase. Where did you hear it? Can you use it in a sentence or a paragraph?
However,one thing that bothers me is that,can we assume u=pi-x and then later assume u=x?
There are two perspectives that lead to the same result.
The semantics you're taught in elementary calculus, all of the occurrences of
x and
u in this problem are dummy variables.
It's not like solving a geometry problem where you have a hypothetical rectangle of and you define
L to mean its length and
W to mean its width and they have a fixed and relevant meaning throughout the lifetime of the problem.
Instead, their only meaning is "I am the variable this definite integral is integrating over" and they have no meaning outside of the integral -- its only purpose is to make it easier to write the function inside the integral. The expressions \int_a^b f(p) \, dp and \int_a^b f(q) \, dq literally mean exactly the same thing. Tnd the notation \int_a^b f is sometimes used when feasible, which further emphasizes the irrelevance of dummy variables.
In the other semantics that's often used (but rarely stated explicitly at your level), the integrals \int_a^b f(p) \, dp and \int_a^b f(q) \, dq might mean different things, but the definition of the integral immediately implies that the two integrals evaluate to the same real number -- you have an identity:
\int_a^b f(p) \, dp = \int_a^b f(q) \, dq
which is just another integral law.
