In the previous posts, we have explored the first part of the Fundamental Theorem of Calculus, which allows us to connect differentiation and integration. Now, we will delve into the second part of the theorem and discover its powerful applications in solving problems involving indefinite integrals and antiderivatives.

To understand the second part of the Fundamental Theorem, it is crucial to grasp the concept of an antiderivative. An antiderivative of a function f(x) is any function F(x) whose derivative is equal to f(x). In other words, if F'(x) = f(x), then F(x) is an antiderivative of f(x).

The second part of the Fundamental Theorem states that if we have an antiderivative F(x) of a function f(x), then the definite integral of f(x) from a to b can be calculated by subtracting the value of F(x) at a from the value of F(x) at b. In mathematical notation, this can be expressed as follows:

∫[a to b] f(x) dx = F(b) - F(a)

Let's consider an example to illustrate this concept. Suppose we want to find the definite integral of the function f(x) = 2x from 1 to 4. First, we find an antiderivative of f(x), which is F(x) = x^2. Using the second part of the Fundamental Theorem, we subtract the value of F(x) at a (1) from the value of F(x) at b (4):

∫[1 to 4] 2x dx = F(4) - F(1) = 16 - 1 = 15.

By using the second part of the Fundamental Theorem, we were able to evaluate the definite integral and find the result in a straightforward manner.

Remember, the second part of the Fundamental Theorem of Calculus provides us with a powerful tool to evaluate definite integrals and work with antiderivatives. By understanding and applying this theorem, you will be able to solve a wide range of problems in calculus. Keep practicing, and always remember that you are capable of mastering this material. Keep up the great work!