Evaluate The Following Integral.$\int \frac{6z^3 + 5z^2 - 68z + 18}{z^2 + Z - 12} \, Dz$

Feb 26, 2025 by ADMIN 89 views

**Evaluating the Given Integral: A Step-by-Step Approach**

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Introduction

In this article, we will delve into the world of calculus and focus on evaluating a given integral. The integral in question is $\int \frac{6z^3 + 5z^2 - 68z + 18}{z^2 + z - 12} \, dz$ . Our goal is to simplify this expression and find its antiderivative. We will break down the solution into manageable steps, making it easier to understand and follow along.

Breaking Down the Integral

To begin, let's examine the integral and identify its components. The numerator is a polynomial of degree 3, while the denominator is a quadratic expression. Our first step will be to factor the denominator, which will allow us to simplify the expression and potentially cancel out common factors.

Factoring the Denominator

The denominator, $z^2 + z - 12$ , can be factored as $(z + 4)(z - 3)$ . This is a difference of squares, where $a^2 - b^2 = (a + b)(a - b)$ . In this case, $a = z$ and $b = 3$ .

import sympy as sp

# Define the variable
z = sp.symbols('z')

# Define the denominator
denominator = z**2 + z - 12

# Factor the denominator
factored_denominator = sp.factor(denominator)

print(factored_denominator)

Simplifying the Integral

Now that we have factored the denominator, we can rewrite the integral as $\int \frac{6z^3 + 5z^2 - 68z + 18}{(z + 4)(z - 3)} \, dz$ . Our next step will be to simplify the numerator by factoring out any common factors.

Factoring the Numerator

The numerator, $6z^3 + 5z^2 - 68z + 18$ , can be factored as $(z + 4)(6z^2 - 7z - 9)$ . This is a product of two binomials, where $a = z + 4$ and $b = 6z^2 - 7z - 9$ .

# Define the numerator
numerator = 6*z**3 + 5*z**2 - 68*z + 18

# Factor the numerator
factored_numerator = sp.factor(numerator)

print(factored_numerator)

Cancelling Common Factors

Now that we have factored both the numerator and the denominator, we can cancel out any common factors. In this case, we can cancel out the factor $(z + 4)$ from both the numerator and the denominator.

Simplified Integral

After cancelling out the common factor, we are left with the simplified integral $\int \frac{6z^2 - 7z - 9}{z - 3} \, dz$ . Our next step will be to evaluate this integral using the method of partial fractions.

Evaluating the Integral using Partial Fractions

To evaluate the integral, we will use the method of partial fractions. This involves expressing the rational function as a sum of simpler fractions, which can then be integrated separately.

Partial Fractions Decomposition

The rational function $\frac{6z^2 - 7z - 9}{z - 3}$ can be decomposed into partial fractions as follows:

\frac{6z^2 - 7z - 9}{z - 3} = \frac{A}{z - 3} + \frac{Bz + C}{z - 3}

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

Finding the Constants

To find the constants $A$ , $B$ , and $C$ , we will multiply both sides of the equation by $(z - 3)$ and then equate the coefficients of the resulting polynomial.

# Define the variables
A, B, C = sp.symbols('A B C')

# Define the equation
equation = sp.Eq(6*z**2 - 7*z - 9, A*(z - 3) + (B*z + C)*(z - 3))

# Solve for the constants
solution = sp.solve(equation, (A, B, C))

print(solution)

Evaluating the Integral

Now that we have found the constants $A$ , $B$ , and $C$ , we can evaluate the integral using the method of partial fractions.

Final Answer

The final answer is $\boxed{2z^2 + 5z + 3\ln|z - 3| + C}$ .

Note: The constant $C$ is an arbitrary constant that can take on any value.

Conclusion

In this article, we evaluated the given integral using the method of partial fractions. We broke down the solution into manageable steps, making it easier to understand and follow along. We factored the denominator, simplified the numerator, cancelled out common factors, and finally evaluated the integral using partial fractions. The final answer is $\boxed{2z^2 + 5z + 3\ln|z - 3| + C}$ .

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Introduction

In our previous article, we evaluated the given integral $\int \frac{6z^3 + 5z^2 - 68z + 18}{z^2 + z - 12} \, dz$ using the method of partial fractions. In this article, we will address some of the common questions and concerns that readers may have.

Q&A Session

Q: What is the method of partial fractions?

A: The method of partial fractions is a technique used to evaluate the integral of a rational function. It involves expressing the rational function as a sum of simpler fractions, which can then be integrated separately.

Q: Why do we need to factor the denominator?

A: Factoring the denominator allows us to simplify the expression and potentially cancel out common factors. This makes it easier to evaluate the integral.

Q: How do we find the constants A, B, and C?

A: To find the constants A, B, and C, we multiply both sides of the equation by (z - 3) and then equate the coefficients of the resulting polynomial.

Q: What is the significance of the constant C?

A: The constant C is an arbitrary constant that can take on any value. It is often referred to as the "constant of integration".

Q: Can we use the method of partial fractions for any rational function?

A: No, the method of partial fractions is only applicable to rational functions that can be expressed as a sum of simpler fractions.

Q: What are some common mistakes to avoid when using the method of partial fractions?

A: Some common mistakes to avoid include:

Not factoring the denominator properly
Not finding the correct values for the constants A, B, and C
Not checking for common factors between the numerator and denominator
Not simplifying the expression properly

Q: Can we use the method of partial fractions for integrals with complex numbers?

A: Yes, the method of partial fractions can be used for integrals with complex numbers. However, the process may be more complicated and require additional steps.

Q: Are there any other methods for evaluating integrals?

A: Yes, there are several other methods for evaluating integrals, including:

Integration by substitution
Integration by parts
Integration by partial fractions
Integration using trigonometric substitution
Integration using hyperbolic substitution

Conclusion

In this article, we addressed some of the common questions and concerns that readers may have when evaluating the given integral using the method of partial fractions. We hope that this Q&A session has been helpful in clarifying any doubts and providing a better understanding of the method.

Additional Resources

For further reading and practice, we recommend the following resources:

Calculus textbooks: There are many excellent calculus textbooks available that cover the method of partial fractions in detail.
Online resources: Websites such as Khan Academy, MIT OpenCourseWare, and Wolfram Alpha offer a wealth of information and practice problems on the method of partial fractions.
Practice problems: We recommend practicing the method of partial fractions with a variety of problems to become more comfortable with the technique.

Final Thoughts

Evaluating integrals using the method of partial fractions can be a challenging but rewarding experience. With practice and patience, you can become proficient in this technique and apply it to a wide range of problems. Remember to always check your work carefully and double-check your answers to ensure accuracy.