First of all I would like to start off by asking why do they have different change of variable formulas for definite integrals than indefinite…why cant we just integrate using U substitution as we normally do in indefinite integral and then sub the original U value back and use that integrand for definite integral?

I was at one point understanding integration but not when they started coming up with different formulas for definite integrals in U-substitution I got lost and resulted to just forcibly memorizing the formulas…

I dont get why for U substitution they sub the upper and lower bounds into U from the original function to find the new upper and lower bounds with the function U.

I know that because if you dont want to sub the original value of U in and want to instead stick to U as your function you must use those new upper and lower bound but if you sub in the original value for U then you can use your old upper and lower bound values.

My question is what or how does plugging your old lower and upper bound values into U give you the new values of your new function thats expressed as U…

Why do they make such a big deal out of it and complicate it when all they have to do is same U sub as indefinite integral and then plug original value of U in and go from there…are these math people just making excuses to come up with more work or is there more logic behind it?