Explain the difference between an indefinite integral and a definite integral.
Choose the best answer below.
A. An indefinite? integral, after evaluating it at the limits of? integration, results in a particular number. A definite integral result in a set of functions that share the same derivative and uses an arbitrary constant of integration.
B. An indefinite integral cannot always be integrated analytically and may require numeric? integration, while it is always possible to integrate a definite integral. Definite integrals always return a real number after evaluation at its limits of integration.
C.A definite? integral, after evaluating it at the limits of?integration, results in a particular number. An indefinite integral result in a set of functions that share the same derivative and uses an arbitrary constant of integration.
D.A definite integral is defined and continuous over the interval of integration and has finite limits of integration. An indefinite integral is also defined and continuous over the interval of? integration but may have plus or minus ± infinity8 as a limit of integration.
The correct answer and explanation is :
The correct answer is:
C. A definite integral, after evaluating it at the limits of integration, results in a particular number. An indefinite integral results in a set of functions that share the same derivative and use an arbitrary constant of integration.
Explanation:
In calculus, integrals are used to find areas under curves, accumulate quantities, and solve differential equations. There are two main types of integrals: indefinite integrals and definite integrals.
Indefinite Integral
An indefinite integral, also known as an antiderivative, represents a family of functions whose derivative is the given function. The result of an indefinite integral includes an arbitrary constant, C, because differentiation removes constant terms.
For example:
[
\int x^2 \, dx = \frac{x^3}{3} + C
]
Since differentiation removes constant terms, the antiderivative includes C to account for all possible original functions.
Definite Integral
A definite integral computes the exact area under the curve of a function between two specific limits of integration, a and b. It is written as:
[
\int_a^b f(x) \, dx
]
The fundamental theorem of calculus states that the definite integral is evaluated by finding the antiderivative of the function and subtracting its values at the upper and lower limits:
[
\int_1^3 x^2 \, dx = \left[ \frac{x^3}{3} \right]_1^3 = \frac{3^3}{3} – \frac{1^3}{3} = \frac{27}{3} – \frac{1}{3} = \frac{26}{3}
]
Unlike an indefinite integral, a definite integral results in a specific numerical value rather than a function with a constant of integration.
Thus, option C correctly describes the difference between these two types of integrals.