That is indeed true for a definite integral, but this insight is considering indefinite integrals, which deals with the problem of finding antiderivatives.When we did integrals, we didn't look "backward" and consider the integral to be "interpreted as all the functions...". Instead, it was interpreted as :the area under the curve of the specified function;
It really depends on people's backgrounds and how recently they've studied all this.I like to think that this has always been known and clear to everybody.
This turns out to be because micromass can do integration by parts in his sleep and I had to think about it a bit :)Can someone explain where the equation comes from that the second paradox starts with? The one where the integrals on both sides cancel leaving 0 = 1?
Just like absolute value of x stands for two (not one) expressions, so does the shortcut C. When I learned this eons ago my teacher made it clear that C stood for two constants, one for each domain x < 0 and x>0.
Start with the idea of an "equivalence relation". This is a relation that is reflexive, symmetric and transitive. Equality is one example of an equivalence relation. Modulus arithmetic is another. Two numbers are "equivalent modulo n" if they each have the same remainder upon division by n. In the case at hand, the equivalence relation is that two functions are equivalent if their first derivatives are identical as functions.Thanks for great insight that even I as an amateur got a lot from.
Can someone explain what the hell this means tho;
[alpha] = 
Yes, you are free to make up any equivalence relationship. It just has to satisfy the properties. It has to be a relation and it has to be reflexive, symmetric and transitive.are you free to make up any equivalence relationship?