Evaluating the Antiderivative

Oct 13, 2025 by recoveryshake.com 104 views

Introduction

Limits and integration are fundamental concepts in calculus that have numerous applications in various fields, including physics, engineering, and economics. In this article, we will explore the concept of limits and integration, focusing on the evaluation of a specific integral. We will examine the limit of a function as the variable approaches a certain value, and then apply this concept to evaluate the given integral.

Limits: A Brief Overview

A limit is a value that a function approaches as the input or independent variable approaches a specific value. In other words, it is the value that the function gets arbitrarily close to as the input gets arbitrarily close to a certain point. Limits are denoted by the symbol $\lim_{x \to a} f(x)$ , where $a$ is the value that the input is approaching, and $f(x)$ is the function being evaluated.

Types of Limits

There are two types of limits: one-sided limits and two-sided limits. A one-sided limit is a limit that approaches a value from one side, either from the left or the right. A two-sided limit is a limit that approaches a value from both sides.

Evaluating Limits

To evaluate a limit, we need to determine the behavior of the function as the input approaches the specified value. There are several techniques for evaluating limits, including:

Direct Substitution: This involves substituting the value of the input into the function and evaluating the result.
Factoring: This involves factoring the function into simpler components and evaluating the limit of each component separately.
L'Hopital's Rule: This involves using the derivative of the function to evaluate the limit.

Integration: A Brief Overview

Integration is the process of finding the area under a curve or the accumulation of a quantity over a given interval. It is denoted by the symbol $\int f(x) \, dx$ , where $f(x)$ is the function being integrated.

Types of Integrals

There are several types of integrals, including:

Definite Integrals: These are integrals that have a specific upper and lower bound.
Indefinite Integrals: These are integrals that do not have a specific upper and lower bound.

Evaluating Integrals

To evaluate an integral, we need to find the antiderivative of the function being integrated. The antiderivative is a function that, when differentiated, gives the original function.

The Given Integral

The given integral is $\int_1^3 \sin \left(x^3+2\right) \, dx$ . This is a definite integral, meaning that it has a specific upper and lower bound.

Evaluating the Integral

To evaluate this integral, we need to find the antiderivative of the function $\sin \left(x^3+2\right)$ . We can do this by using the substitution method.

Substitution Method

The substitution method involves substituting a new variable into the function being integrated. In this case, we can substitute $u = x^3 + 2$ . This gives us $du = 3x^2 \, dx$ .

Evaluating the Antiderivative

We can now evaluate the antiderivative of the function $\sin \left(x^3+2\right)$ . We have:

\int \sin \left(x^3+2\right) \, dx = -\frac{1}{3} \cos \left(x^3+2\right) + C

Applying the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that the definite integral of a function can be evaluated by applying the antiderivative of the function to the upper and lower bounds of the integral.

Evaluating the Definite Integral

We can now evaluate the definite integral $\int_1^3 \sin \left(x^3+2\right) \, dx$ by applying the antiderivative to the upper and lower bounds of the integral.

\int_1^3 \sin \left(x^3+2\right) \, dx = \left[-\frac{1}{3} \cos \left(x^3+2\right)\right]_1^3

Simplifying the Expression

We can now simplify the expression by evaluating the antiderivative at the upper and lower bounds of the integral.

\int_1^3 \sin \left(x^3+2\right) \, dx = -\frac{1}{3} \cos \left(3^3+2\right) + \frac{1}{3} \cos \left(1^3+2\right)

Final Answer

The final answer is:

\int_1^3 \sin \left(x^3+2\right) \, dx = -\frac{1}{3} \cos \left(27+2\right) + \frac{1}{3} \cos \left(1+2\right)

\int_1^3 \sin \left(x^3+2\right) \, dx = -\frac{1}{3} \cos \left(29\right) + \frac{1}{3} \cos \left(3\right)

\int_1^3 \sin \left(x^3+2\right) \, dx = -\frac{1}{3} \left(-0.05234\right) + \frac{1}{3} \left(-0.98999\right)

\int_1^3 \sin \left(x^3+2\right) \, dx = 0.01745 - 0.32999

\int_1^3 \sin \left(x^3+2\right) \, dx = -0.31254

Conclusion

Q&A: Limits and Integration

Q: What is a limit in mathematics?

A: A limit is a value that a function approaches as the input or independent variable approaches a specific value. In other words, it is the value that the function gets arbitrarily close to as the input gets arbitrarily close to a certain point.

Q: What are the different types of limits?

A: There are two types of limits: one-sided limits and two-sided limits. A one-sided limit is a limit that approaches a value from one side, either from the left or the right. A two-sided limit is a limit that approaches a value from both sides.

Q: How do you evaluate a limit?

A: To evaluate a limit, you need to determine the behavior of the function as the input approaches the specified value. There are several techniques for evaluating limits, including direct substitution, factoring, and L'Hopital's rule.

Q: What is integration in mathematics?

A: Integration is the process of finding the area under a curve or the accumulation of a quantity over a given interval. It is denoted by the symbol $\int f(x) \, dx$ , where $f(x)$ is the function being integrated.

Q: What are the different types of integrals?

A: There are several types of integrals, including definite integrals and indefinite integrals. A definite integral has a specific upper and lower bound, while an indefinite integral does not have a specific upper and lower bound.

Q: How do you evaluate an integral?

A: To evaluate an integral, you need to find the antiderivative of the function being integrated. The antiderivative is a function that, when differentiated, gives the original function.

Q: What is the Fundamental Theorem of Calculus?

A: The Fundamental Theorem of Calculus states that the definite integral of a function can be evaluated by applying the antiderivative of the function to the upper and lower bounds of the integral.

Q: How do you apply the Fundamental Theorem of Calculus?

A: To apply the Fundamental Theorem of Calculus, you need to find the antiderivative of the function being integrated and then evaluate it at the upper and lower bounds of the integral.

Q: What is the final answer to the given integral?

A: The final answer to the given integral is -0.31254.

Q: What is the significance of limits and integration in mathematics?

A: Limits and integration are fundamental concepts in mathematics that have numerous applications in various fields, including physics, engineering, and economics. They are used to model real-world phenomena and make predictions about future events.

Q: How do you use limits and integration in real-world applications?

A: Limits and integration are used in a variety of real-world applications, including:

Physics: Limits and integration are used to model the motion of objects and predict their trajectories.
Engineering: Limits and integration are used to design and optimize systems, such as bridges and buildings.
Economics: Limits and integration are used to model economic systems and make predictions about future economic trends.

Conclusion

In this article, we have explored the concept of limits and integration, focusing on the evaluation of a specific integral. We have used the substitution method to evaluate the antiderivative of the function $\sin \left(x^3+2\right)$ , and then applied the Fundamental Theorem of Calculus to evaluate the definite integral. The final answer is -0.31254. We have also answered a variety of questions related to limits and integration, including the definition of a limit, the different types of limits, and the significance of limits and integration in mathematics.