I want to first say thank you for taking a second to look at this. My question is, on my Calc 2 homework, my professor has asked me:
given F(x) = integral of 1/t from 1 to x, x greater than or equal to 1.
First show that that derivative of F(x) (the integral being from 1 to x) = 1/x without referencing the fundamental theorm of calculus. He suggested using the mean value theorm of integration, which makes sense, but I feel like I am just rewriting the proof I looked up, which is bad because it means I don't actuallty understand what is going on. If someone can kind of explain this because my book is kind of vague about it and my professor expects us to just understand.
Second, show that the derivative of F(ax) (the integral from 1 to x) = 1/(x). I understood in Calculus 1 that this could happen since you integrate and then derive which eliminates the constant created, but I guess I can't figure how that works in the proof. Which is bad again since I don't really understand what is happening behind the scenes
Finally, show that F(ax) = F(x) - some constant C, and find the value of that constant. This one I can sort of see is a bridging of the first and second tasks but without understanding them, I can't quite grasp the entire picture. Any explanations of these without using the fundamental theorem of calculus or assuming that the anti-derivative of 1/x = ln x and the reverse, would be much appreciated.
Thank you again for your time,