PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
An indefinite integral is a function that takes the antiderivative of another function. It is visually represented as an integral symbol, a function, and then a dx at the end. The indefinite integral is an easier way to symbolize taking the antiderivative. The indefinite integral is related to the definite integral, but the two are not the same.
I want to talk about the antiderivative relationship and maybe come up with a new notation for antiderivatives. First of all let me remind you that finding all the antiderivatives of functions called antidifferentiation is kind of like when you find the derivative of a function you're differentiating when you find the antiderivatives you're antidifferentiating. Now this relationship is the key capital F prime of x equals little f of x in this relationship you can see that little f is the derivative of big F. You take the derivative of big F you get little f, so little f of x is the derivative of big F of x. So with antiderivatives that relationship is switched big F of x is an antiderivative of little f.
You notice how I use the word an because they're many antiderivatives, and that's what this theorem is about. We also know that if capital F of x is antiderivative of little f of x then capital F of x plus c are all the antiderivatives of little f of x. So not only is there more than one antiderivative but this constitutes all of them. Once you find one antiderivative you add a +c you've got all of them which is nice. Now we have a symbol for the operation of differentiation, this is the symbol right this operator operates on a function and what it gives you is that function's derivative.
We already said that capital F prime is little f so this relationship just reflects that. That means the derivatives of big F of x derivative of big F of x is little f of x there's only one. The operation and antidifferentiation needs a symbol too and this the symbol we give it. It's an elongated s think about the declaration of independence some old document where they used to elongate some of yes's that's what this is it's an s for like summation. The elongated s, the integral is red of f of x dx is capital F of x plus c. It means the antiderivatives of little f of x, the antiderivatives of this guy are these guys. So whatever function you want to find the antiderivatives of you put in this expression and this expression means the antiderivatives of that function.
The expression, the integral of f of x dx is called an indefinite integral and that's what we're going to be studying just remember though when we solve the indefinite integrals we're finding the antiderivatives of the function inside.