Evaluating The Integral Of (7x - 13) / ((3x - 5)(x - 3))

Introduction

In this article, we will delve into the process of evaluating the integral of the rational function ${\frac{7x - 13}{(3x - 5)(x - 3)}}$. This type of integral often appears in calculus and requires the technique of partial fraction decomposition. Partial fraction decomposition is a crucial method for breaking down complex rational functions into simpler fractions, making them easier to integrate. Our primary goal is to express the given integrand, ${\frac{7x - 13}{(3x - 5)(x - 3)}}$, as a sum of simpler fractions, each of which can be integrated using basic integration rules. This method is particularly effective when dealing with rational functions where the denominator can be factored into linear or irreducible quadratic factors. Understanding and mastering this technique is essential for anyone studying calculus, as it is frequently used in various applications, including solving differential equations and evaluating definite integrals. By the end of this discussion, you will have a clear understanding of how to apply partial fraction decomposition to solve integrals of this form, allowing you to tackle similar problems with confidence and precision.

Partial Fraction Decomposition

To begin, we express the integrand as a sum of partial fractions. The given rational function is ${\frac{7x - 13}{(3x - 5)(x - 3)}}$. Since the denominator is a product of distinct linear factors, we can write the partial fraction decomposition in the form:

${ \frac{7x - 13}{(3x - 5)(x - 3)} = \frac{A}{3x - 5} + \frac{B}{x - 3} }$

where A and B are constants that we need to determine. This decomposition is based on the principle that any rational function with a denominator factored into linear terms can be expressed as a sum of fractions, each with one of the linear factors as its denominator and a constant as its numerator. The constants A and B are crucial for correctly breaking down the original complex fraction into simpler, integrable parts. To find these constants, we need to clear the denominators by multiplying both sides of the equation by the common denominator, which in this case is ${(3x - 5)(x - 3)}$. This step is vital as it transforms the equation into a more manageable form, allowing us to solve for A and B using algebraic techniques. Accurately determining A and B is the key to successfully integrating the original rational function, as these values will directly influence the final result of the integration process. By understanding the underlying principles of partial fraction decomposition, we can systematically approach and solve a wide range of integrals involving rational functions.

Multiplying both sides by ${(3x - 5)(x - 3)}$, we get:

${ 7x - 13 = A(x - 3) + B(3x - 5) }$

This equation is the result of eliminating the denominators, making it easier to solve for the unknown constants A and B. The next step involves expanding the right-hand side of the equation, which will allow us to group like terms and form a system of equations. Expanding the equation is a critical step as it sets the stage for equating coefficients, a method used to solve for the constants. Once we expand the equation, we can collect terms involving x and the constant terms separately. This separation is essential because it allows us to create two independent equations: one for the coefficients of x and another for the constant terms. These equations will form a system that can be solved using various algebraic techniques, such as substitution or elimination. By carefully expanding and grouping terms, we ensure that the system of equations we derive accurately reflects the original equation, leading to the correct values for A and B. This systematic approach is a cornerstone of partial fraction decomposition, enabling us to handle more complex rational functions with confidence.

Expanding, we have:

${ 7x - 13 = Ax - 3A + 3Bx - 5B }$

Now, we group the terms with x and the constant terms:

${ 7x - 13 = (A + 3B)x + (-3A - 5B) }$

This grouped equation is the foundation for setting up a system of linear equations, which we will use to solve for A and B. By separating the terms containing x and the constant terms, we've effectively created two distinct equations that relate A and B. Setting up a system of linear equations is a crucial step in the partial fraction decomposition process, as it allows us to translate the problem of finding the constants into a more manageable algebraic problem. The accuracy of these equations is paramount, as they directly influence the values of A and B, and consequently, the final integral. We obtain these equations by equating the coefficients of like terms on both sides of the equation. This method is based on the principle that for two polynomials to be equal, their corresponding coefficients must be equal. By carefully equating these coefficients, we can construct a system of equations that accurately represents the relationships between A and B, paving the way for a straightforward solution using techniques such as substitution or elimination.

Equating the coefficients of x and the constant terms, we get the following system of equations:

${ A + 3B = 7}$ ${ -3A - 5B = -13 }$

This system of equations is the key to unlocking the values of A and B, which are essential for the partial fraction decomposition. To solve this system, we can use various methods, such as substitution, elimination, or matrix methods. Solving this system typically involves manipulating the equations to isolate one variable in terms of the other, or to eliminate one variable altogether. The choice of method often depends on the specific structure of the equations, with the goal of finding the most efficient path to the solution. For instance, we might multiply one or both equations by a constant to make the coefficients of one variable match, allowing us to eliminate that variable by adding or subtracting the equations. Alternatively, we could solve one equation for one variable and substitute that expression into the other equation. The accuracy of the solution is paramount, as any errors in the values of A and B will propagate through the rest of the integration process. Therefore, careful attention to detail and a thorough understanding of algebraic techniques are crucial when solving this system.

We can solve this system by multiplying the first equation by 3 and adding it to the second equation:

${ 3(A + 3B) + (-3A - 5B) = 3(7) + (-13) }$ ${ 3A + 9B - 3A - 5B = 21 - 13 }$ ${ 4B = 8 }$

Thus, B = 2. Substituting B = 2 into the first equation:

${ A + 3(2) = 7}$ ${ A + 6 = 7}$

Thus, A = 1. Now that we have found the values of A and B, we can rewrite the original integral in terms of these simpler fractions. Substituting A = 1 and B = 2 back into the partial fraction decomposition is a critical step, as it allows us to express the original complex rational function as a sum of simpler, more manageable fractions. These fractions are designed to be easily integrable using standard calculus techniques, such as the basic rules for integrating rational functions. The accuracy of these values is paramount, as they directly influence the final result of the integral. A mistake in either A or B will lead to an incorrect integration, highlighting the importance of careful calculation and verification. By correctly substituting these values, we set the stage for the final step of integrating each term separately, leading to the solution of the original integral.

Integrating the Partial Fractions

Now we can rewrite the integral as:

${ \int \frac{7x - 13}{(3x - 5)(x - 3)} dx = \int \left( \frac{1}{3x - 5} + \frac{2}{x - 3} \right) dx }$

This equation represents the original integral expressed as a sum of two simpler integrals, each corresponding to one of the partial fractions we derived earlier. Separating the integral into these simpler components is a key advantage of the partial fraction decomposition technique. Each of these integrals is now in a form that can be easily evaluated using standard integration rules. Specifically, we can use substitution for the first integral and a basic logarithmic rule for both integrals. This step transforms a complex integration problem into a series of more manageable tasks, making the entire process more approachable. By separating the integral, we can focus on each term individually, reducing the likelihood of errors and simplifying the overall calculation. This approach highlights the power of partial fraction decomposition as a tool for simplifying complex integrals into sums of simpler, integrable functions.

We can split this into two separate integrals:

${ \int \frac{1}{3x - 5} dx + \int \frac{2}{x - 3} dx }$

For the first integral, let ${u = 3x - 5}$, so ${du = 3 dx}$, and ${dx = \frac{1}{3} du}$. Thus,

${ \int \frac{1}{3x - 5} dx = \frac{1}{3} \int \frac{1}{u} du = \frac{1}{3} \ln|u| + C_1 = \frac{1}{3} \ln|3x - 5| + C_1 }$

Evaluating each integral separately allows us to apply the appropriate integration techniques to each term, ensuring accuracy and efficiency. For the first integral, we used a u-substitution to simplify the expression, which is a common technique for integrating rational functions where the denominator is a linear expression. This substitution transforms the integral into a standard form that can be easily integrated using the natural logarithm. For the second integral, we can directly apply the logarithmic integration rule, as the denominator is a simple linear factor. By breaking the problem into smaller, more manageable parts, we reduce the complexity of the integration process and minimize the risk of errors. This step-by-step approach is characteristic of effective problem-solving in calculus, where complex tasks are often simplified by breaking them down into their fundamental components.

For the second integral:

${ \int \frac{2}{x - 3} dx = 2 \int \frac{1}{x - 3} dx = 2 \ln|x - 3| + C_2 }$

Combining these results, we get:

${ \int \frac{7x - 13}{(3x - 5)(x - 3)} dx = \frac{1}{3} \ln|3x - 5| + 2 \ln|x - 3| + C }$

where ${C = C_1 + C_2}$ is the constant of integration. Combining the results of the individual integrations is the final step in obtaining the solution to the original integral. We add the results of each integral, along with a single constant of integration, C, which represents the family of antiderivatives for the given function. The constant of integration is a crucial element in indefinite integrals, as it accounts for the fact that the derivative of a constant is zero, meaning there are infinitely many possible antiderivatives that differ only by a constant. The final result, ${\frac{1}{3} \ln|3x - 5| + 2 \ln|x - 3| + C}$, represents the general solution to the integral, encompassing all possible antiderivatives. This comprehensive solution is the culmination of the partial fraction decomposition and integration process, demonstrating the power of these techniques in solving complex calculus problems.

Conclusion

In conclusion, we have successfully evaluated the integral ${\int \frac{7x - 13}{(3x - 5)(x - 3)} dx}$ using the method of partial fraction decomposition. This technique allowed us to break down a complex rational function into simpler fractions that were easier to integrate. The final result is:

${ \int \frac{7x - 13}{(3x - 5)(x - 3)} dx = \frac{1}{3} \ln|3x - 5| + 2 \ln|x - 3| + C }$

where C is the constant of integration. This solution underscores the effectiveness of partial fraction decomposition in tackling integrals of rational functions. Mastering this method is essential for students and professionals in mathematics, engineering, and other fields where calculus is applied. The ability to decompose complex rational functions into simpler components not only simplifies the integration process but also provides a deeper understanding of the underlying mathematical structures. The systematic approach we've demonstrated, from setting up the partial fractions to solving for the constants and integrating each term, can be applied to a wide range of similar problems. This highlights the versatility and importance of partial fraction decomposition as a fundamental tool in integral calculus, enabling the solution of otherwise intractable integrals and providing valuable insights into mathematical problem-solving.