How to use FTC2 to evaluate a definite integral when the antiderivative is known?

The Fundamental Theorem of Calculus (FTC) Part 2 provides a method to evaluate definite integrals when the antiderivative (indefinite integral) of the integrand is known. Here's how you can use FTC2 to evaluate a definite integral:

1. Find the antiderivative: First, you need to find the antiderivative of the integrand. The antiderivative is a function whose derivative is the original function. Let's denote the antiderivative as $F(x)$.

   For example, if you have $\int x^2 \, dx$, the antiderivative $F(x)$ is $\frac{x^3}{3}$.

2. Evaluate $F(x)$ at the limits of integration: Next, evaluate $F(x)$ at the upper limit of integration ($b$) and the lower limit of integration ($a$). This gives you $F(b)$ and $F(a)$.

   Using our example, if the limits of integration are $a=1$ and $b=3$, then $F(3) = \frac{3^3}{3} = 9$ and $F(1) = \frac{1^3}{3} = \frac{1}{3}$.

3. Apply FTC2: According to FTC2, the definite integral of a function $f(x)$ from $a$ to $b$ is given by:

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

   Substitute the values of $F(b)$ and $F(a)$ into this formula:

   $\int_1^3 x^2 \, dx = F(3) - F(1) = 9 - \frac{1}{3} = \frac{26}{3}$