Independence of parametrizations of a line integral

[gotjrgkr]

Homework Statement

Hello!
I've been trying to prove a problem.
I attach the problem. I refer to the book " Advanced calculus" written by watson fulks.
You can find the below information in p. 405~p.417

I want to prove C Exercise problem number 2.

Homework Equations

The Attempt at a Solution

I could prove it when two parametrizations of a curve are bijective. But, I don't have any idea to access how to prove it in general. I even don't know whether the statement in C Exercise 2 is true or not...
Is the statement true??
Then, could you give me a hint??

 

[lippyka]

I think I should use the theorem 8.2 in the book,. but, I can't access how to do it..The statement in C Exercise 2 is false. Theorem 8.2 is a generalization of the Fundamental Theorem of Calculus, and it states that if two parameterizations of a function are related by a differentiable transformation, then the integrals of the two parameterizations are related by the same transformation. In this case, the statement in C Exercise 2 does not hold because the two parameterizations are not related by a differentiable transformation.