In the previous post, we discussed how to find the antiderivative of a function to evaluate definite integrals. Now, we will delve deeper into the second part of the fundamental theorem of calculus, which focuses on evaluating definite integrals using antiderivatives and the limits of integration.

To evaluate a definite integral ∫[a, b] f(x) dx, we can use the following formula:

∫[a, b] f(x) dx = F(b) - F(a)

where F(x) is the antiderivative of f(x). This formula allows us to directly find the exact value of a definite integral without having to evaluate an infinite number of infinitesimal changes.

For example, let's consider the function f(x) = 2x. To find the definite integral of f(x) from 1 to 3, we first need to find the antiderivative of f(x), which is F(x) = x^2 + C, where C is the constant of integration. Applying the formula, we have:

∫[1, 3] 2x dx = F(3) - F(1) = (3^2 + C) - (1^2 + C) = 9 - 1 = 8.

Therefore, the definite integral of f(x) from 1 to 3 is 8.

By using the fundamental theorem of calculus, we can avoid tedious calculations and find the exact value of definite integrals with relative ease.