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Chapter 5: Problem 8

Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?

Short Answer

Expert verified
Answer: The constant of integration can be omitted when evaluating a definite integral because when we subtract the antiderivative at the two limits, a and b, the constants will cancel out. This means that the constant does not affect the final result of the definite integral.

Step by step solution

01

Definition of Antiderivative and Definite Integral

An antiderivative is a function, F(x), whose derivative is equal to a given function, f(x). In other words, F'(x) = f(x). When we find the antiderivative of a function, we often add a constant of integration, C, since differentiation eliminates constants. So, F(x) = 鈭玣(x)dx + C. A definite integral represents the signed area under the curve of a function between two points, a and b. It is denoted as 鈭玔a, b] f(x)dx, and can be evaluated using the Fundamental Theorem of Calculus, which states that if F(x) is an antiderivative of f(x) on the interval [a, b], then 鈭玔a, b] f(x)dx = F(b) - F(a).
02

Explanation for Omitting the Constant of Integration

Since the definite integral is evaluated by finding the difference of the antiderivative at the two limits, a and b, the constant of integration, C, can be omitted. This is because when we subtract F(a) from F(b), the constants will cancel out: F(b) - F(a) = ( 鈭玣(x)dx + C ) - ( 鈭玣(x)dx + C ) = 鈭玣(x)dx - 鈭玣(x)dx As seen in the equation above, the constants, C, cancel out, leaving us with the definite integral 鈭玔a, b] f(x)dx. Therefore, we can omit the constant of integration when evaluating a definite integral, since it will not affect the final result.

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Most popular questions from this chapter

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\sqrt{3}} \frac{3 d x}{9+x^{2}}$$

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