Determine ∫ X X 2 − 6 X + 8 D X \int \frac{x}{x^2-6x+8} \, Dx ∫ X 2 − 6 X + 8 X ​ D X

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Introduction

In this article, we will delve into the world of calculus and focus on determining the integral of a rational function. The given integral is $\int \frac{x}{x^2-6x+8} \, dx$. This type of integral is known as a partial fraction decomposition problem, which involves breaking down a rational function into simpler fractions.

Understanding the Integral

To begin, let's analyze the given integral. We have a rational function of the form $\frac{x}{x^2-6x+8}$. The denominator can be factored as $(x-2)(x-4)$. Therefore, the integral can be rewritten as $\int \frac{x}{(x-2)(x-4)} \, dx$.

Partial Fraction Decomposition

The next step is to perform a partial fraction decomposition on the rational function. This involves expressing the rational function as a sum of simpler fractions. In this case, we can write:

$$\frac{x}{(x-2)(x-4)} = \frac{A}{x-2} + \frac{B}{x-4}$$

where $A$ and $B$ are constants to be determined.

Finding the Constants

To find the values of $A$ and $B$, we can multiply both sides of the equation by the common denominator $(x-2)(x-4)$. This gives us:

$$x = A(x-4) + B(x-2)$$

Now, we can expand the right-hand side of the equation and equate the coefficients of the corresponding terms.

Solving for A and B

Expanding the right-hand side of the equation, we get:

$$x = Ax - 4A + Bx - 2B$$

Equating the coefficients of the corresponding terms, we get:

$$1 = A + B$$

$$0 = -4A - 2B$$

Solving this system of equations, we find that $A = 2$ and $B = -1$.

Rewriting the Integral

Now that we have found the values of $A$ and $B$, we can rewrite the integral as:

$$\int \frac{x}{(x-2)(x-4)} \, dx = \int \frac{2}{x-2} - \frac{1}{x-4} \, dx$$

Evaluating the Integral

To evaluate the integral, we can use the power rule of integration, which states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$. Applying this rule to each term, we get:

$$\int \frac{2}{x-2} - \frac{1}{x-4} \, dx = 2\ln|x-2| - \ln|x-4| + C$$

Conclusion

In conclusion, we have determined the integral of the rational function $\frac{x}{x^2-6x+8}$. The integral can be expressed as $2\ln|x-2| - \ln|x-4| + C$. This result can be verified by differentiating the integral and checking that it equals the original function.

Final Answer

The final answer is $\boxed{2\ln|x-2| - \ln|x-4| + C}$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Factor the denominator of the rational function.
2. Perform a partial fraction decomposition on the rational function.
3. Find the values of the constants $A$ and $B$.
4. Rewrite the integral using the partial fraction decomposition.
5. Evaluate the integral using the power rule of integration.
6. Simplify the result and add the constant of integration.

Tips and Tricks

Here are some tips and tricks to help you solve this type of problem:

• Make sure to factor the denominator of the rational function before performing the partial fraction decomposition.
• Use the power rule of integration to evaluate the integral.
• Simplify the result and add the constant of integration.
• Check your work by differentiating the integral and checking that it equals the original function.

Common Mistakes

Here are some common mistakes to avoid when solving this type of problem:

• Failing to factor the denominator of the rational function.
• Making mistakes when finding the values of the constants $A$ and $B$.
• Failing to simplify the result and add the constant of integration.
• Not checking your work by differentiating the integral and checking that it equals the original function.

Real-World Applications

This type of problem has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, the integral of a rational function can be used to model the motion of an object under the influence of a force. In engineering, the integral of a rational function can be used to design and optimize systems such as electrical circuits and mechanical systems. In economics, the integral of a rational function can be used to model the behavior of economic systems and make predictions about future trends.

Conclusion

In conclusion, determining the integral of a rational function is an important problem in calculus that has many real-world applications. By following the steps outlined in this article, you can solve this type of problem and gain a deeper understanding of the underlying mathematics.

Introduction

In our previous article, we discussed how to determine the integral of a rational function. In this article, we will answer some common questions that students often have when working with rational functions and their integrals.

Q: What is a rational function?

A: A rational function is a function that can be expressed as the ratio of two polynomials. In other words, it is a function of the form $\frac{f(x)}{g(x)}$, where $f(x)$ and $g(x)$ are polynomials.

Q: How do I determine the integral of a rational function?

A: To determine the integral of a rational function, you need to follow these steps:

1. Factor the denominator of the rational function.
2. Perform a partial fraction decomposition on the rational function.
3. Find the values of the constants $A$ and $B$.
4. Rewrite the integral using the partial fraction decomposition.
5. Evaluate the integral using the power rule of integration.
6. Simplify the result and add the constant of integration.

Q: What is partial fraction decomposition?

A: Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. It involves breaking down a rational function into a sum of fractions with linear or quadratic denominators.

Q: How do I find the values of the constants $A$ and $B$?

A: To find the values of the constants $A$ and $B$, you need to equate the coefficients of the corresponding terms on both sides of the equation. This will give you a system of equations that you can solve to find the values of $A$ and $B$.

Q: What is the power rule of integration?

A: The power rule of integration is a rule that states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$. This rule can be used to evaluate the integral of a rational function.

Q: How do I simplify the result and add the constant of integration?

A: To simplify the result, you need to combine like terms and simplify the expression. To add the constant of integration, you need to add a constant term to the result.

Q: What are some common mistakes to avoid when determining the integral of a rational function?

A: Some common mistakes to avoid when determining the integral of a rational function include:

• Failing to factor the denominator of the rational function.
• Making mistakes when finding the values of the constants $A$ and $B$.
• Failing to simplify the result and add the constant of integration.
• Not checking your work by differentiating the integral and checking that it equals the original function.

Q: What are some real-world applications of determining the integral of a rational function?

A: Determining the integral of a rational function has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, the integral of a rational function can be used to model the motion of an object under the influence of a force. In engineering, the integral of a rational function can be used to design and optimize systems such as electrical circuits and mechanical systems. In economics, the integral of a rational function can be used to model the behavior of economic systems and make predictions about future trends.

Q: How can I practice determining the integral of a rational function?

A: You can practice determining the integral of a rational function by working through examples and exercises. You can also use online resources such as calculators and software to help you practice.

Conclusion

In conclusion, determining the integral of a rational function is an important problem in calculus that has many real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can determine the integral of a rational function and gain a deeper understanding of the underlying mathematics.

Final Tips

Here are some final tips to help you determine the integral of a rational function:

• Make sure to factor the denominator of the rational function before performing the partial fraction decomposition.
• Use the power rule of integration to evaluate the integral.
• Simplify the result and add the constant of integration.
• Check your work by differentiating the integral and checking that it equals the original function.
• Practice, practice, practice! The more you practice, the more comfortable you will become with determining the integral of a rational function.