The fundamental theorem of calculus

The fundamental theorem of calculus is the most important theorem in calculus and is named very appropriately since it establishes a relationship between differential calculus and integral calculus. Let's see how.

Suppose that f(x) is continuous on [a, b] and differentiable at (a, b), and that F(x) is the antiderivative of f(x). Then, we have the following:

Let's rewrite the preceding equation a bit so it becomes this equation:

All we have done here is replace x with t and b with x. And we know that F(x)-F(a) is also a function. From this, we can derive the following property:

We can derive the preceding property since F(a) is a constant and thus has the derivative zero.

By shifting our point of view a bit, we get the following function:

Therefore, we get .

In summary, if we integrate our function f and then differentiate it, we end up with the original function f.