Basically an integral computes the area between a function and a coordinate axis.
x=5 is a perfectly good function. Its integral is 5x. If you evaluate this between the origin and say 5, the answer is 25 or the area of a square with side 5. If you attempt to find the area of a rectangle which has an infinite side you will also find it has an infinite area.
The derivative of a function gives the slope at any point.
These are really way to big of questions to be answered on a Internet forum, you need to refer to a calculus text to get the best information.
Here are some basic about the integral that should help your understanding:
1) The integral symbol is actually the letter “S” for “SUM” that has been elongated.
2) Q: If it is a “SUM” what is it that I’m adding?
A: What you are adding is an infinite number of dx, dy, dz, etc. where dx, dy, dz are infinitely small. So, if you add dx1+dx2+dx3+dx4+….. to infinity you should get “X” thus integral dx = X ( plus the constant of integration).
3) The integral operates on the differential (dx, dy, dx) and not on numbers. Unless these are in your equations, the problem is undefined.
Integral 5 is undefined
Integral X is undefined
Integral 5X is undefined
Integral 5dx is defined
Integral Xdx is defined
Integral 5Xdy is defined
4) The integration is defined only on the Real and Imaginary number systems (as far as I know) and is not defined on the Natural, Integer, etc. This is because the Natural, Integer number systems do not have the density property and continuous functions cannot be defined there. Remember every Natural number is also a member of the Real number system.
that's what it is, you re sampling f(x) at integer values of x, and multiplying each value by the distance between samples, 1, to give you the area under a step-approximation to the function f(x).
Now think about SUM(i=0, 0.5, 1, 1.5, ..., n-0.5, n) of [f(x)*0.5]
That is a better approximation to the area under the curve, because you're using twice the number of samples. The integral is the limit of this to infinity.
INT(from 0 to n) of f(x) dx = limit as δx-->0 of SUM(i=0, δx, 2δx, ... n-δx, n) of [f(x)*δx]
Basically you're adding an infinite number of infinitely thin blocks, to get an exact value for the area underneath. And the principles of calculus shows that the integral of f(x)*dx is the antiderivative of f(x), so you use the same rules as differentiation but backwards.
You can write down any function to integrate, but in some cases it won't make sense. In the case you suggest, f(x) = 5, it is simple and the area underneath is infinite for your stated domain.
The derivative is the slope of the tangent line to the curve.