Chapter 17: Problem 2

By expressing the following in partial fractions evaluate the given integral. Remember to select the correct form for the partial fractions. $$ \int \frac{13 x-4}{6 x^{2}-x-2} \mathrm{~d} x $$

Short Answer

Expert verified

Question: Evaluate the integral of the following rational function: $$ \int \frac{13x - 4}{6x^2 - x - 2} dx $$ Answer: The evaluated integral is $$ \int \frac{13x - 4}{(2x - 1)(3x+2)} dx = \ln\left|\frac{2x-1}{3x+2}\right| + C $$ where C is the constant of integration.

Step by step solution

Perform partial fraction decomposition

First, we need to factor the denominator of the given rational function, which is: $$ 6x^2 - x - 2 $$ Upon factoring, we get: $$ (2x - 1)(3x + 2) $$ Now, we can express our given rational function as the sum of two simpler fractions: $$ \frac{13x - 4}{(2x - 1)(3x + 2)} = \frac{A}{2x - 1} + \frac{B}{3x + 2} $$ To solve for A and B, we can eliminate the denominators by multiplying both sides by (2x - 1)(3x + 2). This gives us: $$ 13x - 4 = A(3x + 2) + B(2x - 1) $$

Solve for A and B

Let's solve for A and B by using the following two equations (by plugging in appropriate values of x that make either A or B term go to 0): 1. For x = 1/2, we obtain: $$ 13 (1/2) - 4 = A(3(1/2) + 2) \Rightarrow A = 2 $$ 2. For x = -2/3, we obtain: $$ 13 (-2/3) - 4 = B(2(-2/3) - 1) \Rightarrow B = -3 $$ Now we have found A and B: $$ A = 2, \qquad B = -3 $$ This allows us to rewrite the rational function: $$ \frac{13x - 4}{(2x - 1)(3x+2)} = \frac{2}{2x-1} - \frac{3}{3x+2} $$

Integrate the partial fractions

Now that we have decomposed the given rational function into partial fractions, we can integrate each term with respect to x: $$ \int \frac{13x - 4}{(2x - 1)(3x+2)} dx = \int \frac{2}{2x-1} dx - \int \frac{3}{3x+2} dx $$ These integrals can be easily evaluated thanks to their simple form: $$ \int \frac{2}{2x-1} dx = 2\int \frac{dx}{2x-1} = \ln|2x-1|+C_1 $$ $$ \int \frac{3}{3x+2} dx =3 \int \frac{dx}{3x+2} = \ln|3x+2|+C_2 $$

Combine the results

Finally, we can combine the two integrals we obtained above to get the final answer: $$ \int \frac{13x - 4}{(2x - 1)(3x+2)} dx = \ln|2x-1| - \ln|3x+2| + C $$ So, the evaluated integral is $$ \int \frac{13x - 4}{(2x - 1)(3x+2)} dx = \ln\left|\frac{2x-1}{3x+2}\right| + C $$ where C is the constant of integration.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integrating Rational Functions

Integrating rational functions involves finding the antiderivative of a fraction where both the numerator and the denominator are polynomials. A common method to integrate such functions is partial fraction decomposition, which simplifies complex fractions into simpler, easier-to-integrate components.

In our exercise, we start with a rational function $\frac{13x - 4}{6x^2 - x - 2}$ and apply partial fraction decomposition to express it as a sum of simpler fractions. Each of these can then be integrated using basic integration rules. This process often reveals logarithmic functions in the result, as in the case of $\ln|2x-1|$ and $\ln|3x+2|$.

Mathematical Integration Techniques

Various mathematical integration techniques can be employed to tackle different types of integrals. Aside from partial fraction decomposition, other methods include substitution, integration by parts, and trigonometric integration, among others.

Effective use of these techniques often requires recognizing patterns and applying specific strategies that correspond to the form of the integral. In our exercise, identifying that the denominator could be factored was crucial, as it allowed the application of partial fraction decomposition, turning a seemingly complex integral into a series of simpler logarithmic integrals.

Factorization in Algebra

Factorization is a pivotal skill in algebra, involving the breaking down of complex expressions into products of simpler factors. This is particularly useful in partial fraction decomposition, where the denominator of the rational expression must be factored before the expression can be decomposed.

In the given exercise, the quadratic denominator $6x^2 - x - 2$ is factored into $2x - 1$ and $3x + 2$, enabling the decomposition. Mastery of factorization techniques like looking for common factors, applying the difference of squares, and using the quadratic formula can significantly ease the process of integrating rational functions.

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

By expressing the following in partial fractions evaluate the given integral. Remember to select the correct form for the partial fractions. $$ \int \frac{13 x-4}{6 x^{2}-x-2} \mathrm{~d} x $$

Short Answer

Step by step solution

Perform partial fraction decomposition

Solve for A and B

Integrate the partial fractions

Combine the results

Key Concepts

Integrating Rational Functions

Mathematical Integration Techniques

Factorization in Algebra

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity

Force

Magnetism and Electromagnetic Induction

Electromagnetism

Turning Points in Physics

Solid State Physics

Study anywhere. Anytime. Across all devices.