What is the Second Fundamental Theorem of Calculus (FTC2) and how does it help evaluate definite integrals?

The Second Fundamental Theorem of Calculus (FTC2) is a crucial result that establishes a relationship between differentiation (differential calculus) and integration (integral calculus). It provides a way to evaluate definite integrals using antiderivatives, making it a powerful tool in calculus. Here's how it works:

FTC2 Statement: If a function f(x) is continuous on the interval [a, b], and F(x) is a function defined on the same interval such that F'(x) = f(x) for all x in (a, b), then the definite integral of f(x) from a to b is equal to F(b) - F(a). In other words,

$\int_a^b f(x) \, dx = F(b) - F(a)$

How FTC2 helps evaluate definite integrals:

1. Find the antiderivative (indefinite integral): First, you need to find a function F(x) whose derivative is f(x). This is the antiderivative of f(x).

2. Evaluate at the limits of integration: Once you have F(x), evaluate it at the upper limit b and the lower limit a.

3. Subtract the results: Subtract the value of F(x) at a from the value of F(x) at b.

4. Apply FTC2: The result of this subtraction is the definite integral of f(x) from a to b. That is,

   $\int_a^b f(x) \, dx = F(b) - F(a)$

This theorem allows us to evaluate definite integrals without having to use the fundamental building blocks of integration, such as Riemann sums or limits of sums. Instead, we can use antiderivatives, making the process much simpler and more straightforward.