Mastering the Fundamental Theorem of Calculus: Evaluating the Expression ∫f(x) dx – ∫g(x) dx with Limits of Integration

∫f(x) dx – ∫g(x) dx

∫[f(x) – g(x)] dx

The expression ∫f(x) dx – ∫g(x) dx is a combination of two definite integrals with functions f(x) and g(x), respectively. To evaluate this expression, we need to apply the fundamental theorem of calculus, which states that the definite integral of a function can be evaluated by finding its antiderivative and evaluating it at the limits of integration.

1. Find the antiderivatives of f(x) and g(x)

The antiderivative of f(x) is denoted as F(x), and the antiderivative of g(x) is denoted as G(x). We can find these antiderivatives by integrating their respective functions:

F(x) = ∫f(x) dx

G(x) = ∫g(x) dx

2. Evaluate the expression using the limits of integration

Now we can evaluate the expression using the limits of integration. Let a and b be the lower and upper limits of integration, respectively.

∫f(x) dx – ∫g(x) dx = [F(x)]_a^b – [G(x)]_a^b

Plugging in the limits of integration, we get:

[F(b) – F(a)] – [G(b) – G(a)]

Simplifying this expression, we get:

F(b) – G(b) – F(a) + G(a)

This is the final answer for the expression ∫f(x) dx – ∫g(x) dx, evaluated at the limits of integration [a, b].

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