How to Calculate the Integral from a to c of f(x) with Respect to x using Antiderivatives

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To integrate the function f(x) from a to c with respect to x, we can evaluate the antiderivative of f(x) and calculate the difference at the limits a and c

To integrate the function f(x) from a to c with respect to x, we can evaluate the antiderivative of f(x) and calculate the difference at the limits a and c.

The antiderivative of a function f(x) is denoted by F(x) such that F'(x) = f(x).

So, the integral from a to c of f(x) with respect to x is equal to F(c) – F(a).

Let’s go through an example to illustrate this concept:

Suppose we have the function f(x) = x^2 and the limits a = 2 and c = 5.

First, we find the antiderivative of f(x) by integrating the function:
F(x) = (1/3)x^3. (The antiderivative of x^2 is (1/3)x^3 + C, where C is the constant of integration).

Now, we can evaluate the integral from a to c:
∫ (from a to c) f(x) dx = F(c) – F(a)
= [(1/3)(5)^3] – [(1/3)(2)^3]
= (1/3)(125) – (1/3)(8)
= (125/3) – (8/3)
= 117/3.

Therefore, the integral from a to c of f(x) with respect to x is equal to 117/3 for the given function and limits.

It’s important to note that this method applies to any continuous function f(x) that is integrable over the interval [a, c].

More Answers:

Understanding the Definite Integral: A Fundamental Concept in Calculus and How to Compute it
Mastering the Indefinite Integral: Exploring the Fundamental Concept of Antiderivative in Calculus
Understanding the Fundamentals: Evaluating the Integral with Same Limits of Integration

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