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An **antiderivative** of a function is another function whose derivative is the original function. Think of it as reversing the process of differentiation. For example, if we have a function like \(f(x) = -2 e^{3x}\), finding the antiderivative means finding a function F(x) such that \(F'(x) = f(x)\). Here is how to find it:

If we consider the original function \( -2 e^{3x}\), its antiderivative is \( -\frac{2}{3} e^{3x}\). This is because when we differentiate \( -\frac{2}{3} e^{3x}\) with respect to x, the chain rule gives us back \( -2 e^{3x}\).

So, knowing how to find the antiderivative is crucial for solving definite integrals, because it lets us express the accumulated change, as we see in the given exercise.

The **Fundamental Theorem of Calculus** links the concept of differentiation with that of integration. It has two main parts:

• First part: If F is an antiderivative of f on an interval [a, b], then \(\int_{a}^{b} f(x) \, dx = F(b) - F(a)\).
• Second part: If f is continuous on [a, b], then the function F defined by \(F(x) = \int_{a}^{x} f(t) \, dt\) for x in [a, b] is an antiderivative of f.

In our exercise, the Fundamental Theorem of Calculus lets us evaluate the definite integral quickly by using the antiderivative of the integrand. We found that the antiderivative of \( -2 e^{3x}\) is \( -\frac{2}{3} e^{3x}\), and then we applied the limits of integration from 0 to b. This gives us:

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