Mastering the Fundamental Theorem of Calculus: How to Solve Integrals using the Antiderivative Method

(integral a-. b) of f'(x)

To solve the integral of f'(x) from a to b, we can use the fundamental theorem of calculus

To solve the integral of f'(x) from a to b, we can use the fundamental theorem of calculus. According to this theorem, if F(x) is the integral of f(x) and f'(x) is the derivative of F(x), then the integral of f'(x) from a to b is equal to F(b) – F(a).

In simpler terms, we need to find the antiderivative of f(x), represented as F(x), and then evaluate F(b) and F(a) to find the final result.

Here are the steps:

1. Find the antiderivative F(x) of f'(x). This means finding a function whose derivative is equal to f'(x). For example, if f'(x) = 3x^2, then F(x) would be x^3 + C, where C is the constant of integration.

2. Evaluate F(b) and F(a). Substitute the values of b and a into the antiderivative F(x) obtained in step 1. This will give you two values, F(b) and F(a).

3. Subtract F(a) from F(b). Take the difference between F(b) and F(a) to get the result of the integral. This will involve subtracting two values obtained from step 2.

Therefore, the integral of f'(x) from a to b is equal to F(b) – F(a).

It’s important to note that the constant of integration (C) should be added when finding the antiderivative F(x). Also, ensure that the function f(x) is differentiable and continuous in the interval [a, b] for this theorem to be applicable.

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