Use the power rule of integration

Oct 13, 2025 by leolavanderia.com.br 73 views

Introduction

Calculus is a branch of mathematics that deals with the study of continuous change, particularly in the context of functions and limits. One of the fundamental concepts in calculus is integration, which is used to find the area under curves, volumes of solids, and other quantities. In this article, we will evaluate the integral $\int_1^4 -x^{-2} \, dx$ , which is a classic example of a definite integral.

Understanding the Integral

The given integral is $\int_1^4 -x^{-2} \, dx$ . To evaluate this integral, we need to understand the concept of indefinite integrals. An indefinite integral is a function that is the antiderivative of a given function. In other words, it is a function that, when differentiated, gives the original function.

The given integral can be written as $\int -x^{-2} \, dx$ . To evaluate this integral, we can use the power rule of integration, which states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ , where $n$ is a constant.

Evaluating the Integral

Using the power rule of integration, we can evaluate the integral as follows:

$\int -x^{-2} \, dx = -\int x^{-2} \, dx$

$= -\frac{x^{-1}}{-1} + C$

$= x^{-1} + C$

$= \frac{1}{x} + C$

where $C$ is the constant of integration.

Applying the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that the definite integral of a function $f(x)$ from $a$ to $b$ is equal to the antiderivative of $f(x)$ evaluated at $b$ minus the antiderivative of $f(x)$ evaluated at $a$ . In other words, $\int_a^b f(x) \, dx = F(b) - F(a)$ , where $F(x)$ is the antiderivative of $f(x)$ .

In this case, we have $\int_1^4 -x^{-2} \, dx$ . Using the antiderivative we found earlier, we can evaluate this integral as follows:

$\int_1^4 -x^{-2} \, dx = \left[ \frac{1}{x} \right]_1^4$

$= \frac{1}{4} - \frac{1}{1}$

$= \frac{1}{4} - 1$

$= -\frac{3}{4}$

Conclusion

In this article, we evaluated the integral $\int_1^4 -x^{-2} \, dx$ using the power rule of integration and the Fundamental Theorem of Calculus. We found that the antiderivative of $-x^{-2}$ is $\frac{1}{x}$ , and using this antiderivative, we evaluated the definite integral to be $-\frac{3}{4}$ .

Applications of Integration

Integration has numerous applications in various fields, including:

Physics: Integration is used to find the position, velocity, and acceleration of objects under the influence of forces.
Engineering: Integration is used to design and optimize systems, such as electrical circuits, mechanical systems, and control systems.
Economics: Integration is used to model economic systems, including supply and demand curves, and to optimize economic decisions.
Computer Science: Integration is used in computer graphics, game development, and scientific simulations.

Tips and Tricks

Use the power rule of integration to evaluate integrals of the form $\int x^n \, dx$ .
Use the Fundamental Theorem of Calculus to evaluate definite integrals.
Check your work by differentiating the antiderivative to ensure that it matches the original function.
Practice, practice, practice to become proficient in evaluating integrals.

References

Spivak, M. (2008). Calculus. Cambridge University Press.
Stewart, J. (2012). Calculus: Early Transcendentals. Cengage Learning.
Khan Academy. (n.d.). Calculus. Retrieved from https://www.khanacademy.org/math/calculus
MIT OpenCourseWare. (n.d.). Calculus. Retrieved from https://ocw.mit.edu/courses/mathematics/18-01-calculus-i-fall-2007/

Introduction

In our previous article, we evaluated the integral $\int_1^4 -x^{-2} \, dx$ using the power rule of integration and the Fundamental Theorem of Calculus. In this article, we will answer some common questions related to this integral.

Q: What is the antiderivative of $-x^{-2}$ ?

A: The antiderivative of $-x^{-2}$ is $\frac{1}{x}$ .

Q: How do I evaluate the definite integral $\int_1^4 -x^{-2} \, dx$ ?

A: To evaluate the definite integral, you can use the Fundamental Theorem of Calculus, which states that the definite integral of a function $f(x)$ from $a$ to $b$ is equal to the antiderivative of $f(x)$ evaluated at $b$ minus the antiderivative of $f(x)$ evaluated at $a$ . In this case, we have $\int_1^4 -x^{-2} \, dx = \left[ \frac{1}{x} \right]_1^4 = \frac{1}{4} - \frac{1}{1} = -\frac{3}{4}$ .

Q: What is the significance of the constant of integration $C$ ?

A: The constant of integration $C$ is a constant that is added to the antiderivative to make it exact. In other words, it is a constant that is added to the antiderivative to make it equal to the original function.

Q: Can I use the power rule of integration to evaluate integrals of the form $\int x^{-n} \, dx$ ?

A: Yes, you can use the power rule of integration to evaluate integrals of the form $\int x^{-n} \, dx$ . The power rule of integration states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ , where $n$ is a constant.

Q: How do I check my work when evaluating an integral?

A: To check your work, you can differentiate the antiderivative to ensure that it matches the original function. In other words, you can take the derivative of the antiderivative and compare it to the original function to ensure that they are equal.

Q: What are some common mistakes to avoid when evaluating integrals?

A: Some common mistakes to avoid when evaluating integrals include:

Forgetting to add the constant of integration $C$ : Make sure to add the constant of integration $C$ to the antiderivative to make it exact.
Not using the correct antiderivative: Make sure to use the correct antiderivative when evaluating an integral.
Not checking your work: Make sure to check your work by differentiating the antiderivative to ensure that it matches the original function.

Q: What are some advanced topics in integration?

A: Some advanced topics in integration include:

Integration by parts: This is a technique used to evaluate integrals of the form $\int f(x)g(x) \, dx$ .
Integration by substitution: This is a technique used to evaluate integrals of the form $\int f(g(x)) \, dx$ .
Improper integrals: This is a technique used to evaluate integrals that have infinite limits of integration.

Q: Where can I find additional practice problems and explanations?

A: You can find additional practice problems and explanations on websites such as Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.

Q: What are some resources for learning more about integration?

A: Some resources for learning more about integration include:

Calculus textbooks: For a comprehensive introduction to calculus, including integration, see "Calculus" by Michael Spivak or "Calculus: Early Transcendentals" by James Stewart.
Online resources: For additional practice problems and explanations, see Khan Academy's calculus course or MIT OpenCourseWare's calculus course.
Research papers: For advanced topics in integration, see research papers on arXiv or MathSciNet.

Conclusion

In this article, we answered some common questions related to the integral $\int_1^4 -x^{-2} \, dx$ . We hope that this article has been helpful in clarifying any confusion and providing additional practice problems and explanations.