how did they get from the first line to the second line (1 Viewer)

[chilli 412]

the implication from this is that (bounds of integration are unchanged so both integrands must be equal)
however an investigation on desmos shows this is not at all the case

what magic is going on here??

 

[gazzaboy]

Just because the definite integrals are the same, doesn't mean the functions themselves are the same e.g. integrating f(x) = x and g(x) = 1-x between 0 and 1 give the same answer, but are different functions.

I don't think it's saying that you can easily go from the first line to the second line. I think it's just saying that both of those integrals compute to equal to K. I don't think it's easy to show the second integral equals to Catalan's constant.

 

[chilli 412]

Just because the definite integrals are the same, doesn't mean the functions themselves are the same e.g. integrating f(x) = x and g(x) = 1-x between 0 and 1 give the same answer, but are different functions.

I don't think it's saying that you can easily go from the first line to the second line. I think it's just saying that both of those integrals compute to equal to K. I don't think it's easy to show the second integral equals to Catalan's constant.

i see, i dont know why i never considered that the functions could be different, thanks for the help (p.s i wonder if there is some kind of shared properties that g(x) and f(x) have that allow their areas under the graph from a to b to be equal even though g(x) != f(x), or if it is just a matter of chance)

 

[gazzaboy]

Unfortunately there aren't too many other shared properties of functions with the same definite integral across the same interval. For example, if we take the function defined by f(x) = x and find the integral between 0 and 1, the value is 1/2. But we could find an infinite number of functions which have the same definite integral in this interval [0,1].

Note: If however you can say that two continuous functions f and g have the same definite integral in the interval [a,b] for any a,b, then you have say that f and g are equal by the fundamental theorem of calculus.