Almost every engineer or scientist needs to make a hard integration one day, exact equation integration or integration exact formula, sometimes the exact equation solution out which the exact equation solver, is not easy sometimes to figure out also it and the natural opposite the exact solution of differential equation out which the exact equation differential solver in calculus both obvious which to make the profile and the behavior of functions in functional profile analysis play the important place in math rather advanced or higher mathematics or not.
———————————————————
Functions versus variables is one wonderful relationship in mathematics, a variable sometimes independent, beside it plays some specials in independent variables statistics it almost essential, let us define it x in here, or sometimes dependent, you find more about it in dependent variable definition, let us define it y or Y in here, y depends on x and x does not depends on y in the relationship y=f(x), sometimes we go and plot a profile and sometimes go and make a table.
———————————————————
When one wants to integrate f(x)=y it ends up with Y=F(x)+constant, we assume that the integration was could have been to carry out, if it is not we can say he or she is in trouble, and we call Y is the exact solution, it needs a boundary condition (Y,x) though to figure out the constant, you will find more in boundary condition types, x is an independent variable here and both y and Y are dependents, we are not finished yet, in case it was not could have been to carry out the required integration some go and imaging the interval of x and make an integration with two limits which is the fundamental, this is numerically with the computer sometime, everything is OK.
———————————————————
Now why do we need iteration, they’re many iteration algorithms and some succeeded to make iterative usually with iteration iterative variables, this simply because y is function of the independent x and of the dependent y, i. e. depend on itself equation though, which x and y satisfy y=f(x,y), almost self behavior equation formula, more what if we need to integrate , sometimes making things take place and happen.