Question

Use the method of partial fractions to evaluate each of the following integrals. \(\int \frac{3 x}{x^{2}+2 x-8} d x\)

Short answer

\(2\ln|x+4| + \frac{3}{2}\ln|x-2| + C\)

Step-by-step solution

Factor the Denominator

First, we need to factor the quadratic expression in the denominator, \(x^2 + 2x - 8\). This can be factored into \((x + 4)(x - 2)\).

Set Up Partial Fractions

Write the expression \(\frac{3x}{x^2 + 2x - 8}\) as a sum of partial fractions. Since the denominator is \((x + 4)(x - 2)\), set up \(\frac{3x}{(x+4)(x-2)} = \frac{A}{x+4} + \frac{B}{x-2}\).

Solve for Coefficients

Multiply both sides by the common denominator \((x+4)(x-2)\) to get \(3x = A(x-2) + B(x+4)\). To find \(A\) and \(B\), substitute suitable values for \(x\):- Let \(x = -4\), then \(3(-4) = A(-4-2) + B(0)\), solving gives \(A = 2\).- Let \(x = 2\), then \(3(2) = A(0) + B(2+4)\), solving gives \(B = \frac{3}{2}\).

Rewrite the Integral

With \(A = 2\) and \(B = \frac{3}{2}\), rewrite the integral as the sum of integrals:\[\int \frac{2}{x+4} \,dx + \int \frac{3/2}{x-2} \,dx\]

Integrate Each Term

Integrate each term separately:- The integral of \(\frac{2}{x+4}\) is \(2\ln|x+4| + C_1\).- The integral of \(\frac{3/2}{x-2}\) is \(\frac{3}{2}\ln|x-2| + C_2\).

Combine the Results

Combine the integrated terms to get the final solution:\[2\ln|x+4| + \frac{3}{2}\ln|x-2| + C\] where \(C = C_1 + C_2\) is the constant of integration.

Key concepts

Integration Techniques

Integration techniques are methods used to compute integrals, which are fundamental in calculus for finding the area under curves. One powerful technique is the method of partial fractions, especially useful for integrating rational functions. Rational functions are fractions where both the numerator and the denominator are polynomials.

In this problem, our goal is to integrate the function \(\frac{3x}{x^2 + 2x - 8}\). When dealing with such functions, partial fractions allow us to split the complex fraction into simpler ones that are easier to integrate.

The technique involves these steps:

• Factor the denominator to find the simpler components.
• Express the original fraction as a sum of simpler fractions based on these components.
• Solve for the constants by equating coefficients.
• Integrate each simpler fraction separately.
• Combine the results to find the complete integral.

This process transforms a challenging integral into a series of basic ones, making it a cornerstone method in calculus.

Calculus

Calculus is the branch of mathematics that studies change and motion. It is divided into two main areas: differentiation (dealing with rates of change and slopes) and integration (concerned with accumulation of quantities and areas under curves).

In the world of calculus, integration plays a critical role. It helps you find solutions to problems involving areas, volumes, central points, and many other crucial calculations. The integral of a function gives the total accumulation of that function, providing insights into how the function behaves over a certain interval.

For this specific exercise, the integral \(\int \frac{3x}{x^2 + 2x - 8}\, dx\) uses integration to find the accumulated value over a function expressed by a rational polynomial. Once the integral is solved, it not only provides us with a solution but also demonstrates the power and utility of calculus in solving real-world problems.

From engineering to physics, calculus and integration techniques like partial fractions underpin numerous applications, showcasing their importance in various fields.

Polynomial Fractions

Polynomial fractions are expressions where the numerator and the denominator are polynomials. These are common in mathematics, especially when dealing with rational functions. They appear frequently in calculus problems and require special techniques for integration and differentiation.

To handle polynomial fractions, it often involves first simplifying the expression. In our example of \(\frac{3x}{x^2+2x-8}\), the denominator \(x^2+2x-8\) is a quadratic polynomial that can be factored into components, \((x + 4)(x - 2)\). Each component represents a simpler element we can work with through partial fractions.

This simplification is crucial because it allows us to express the complex polynomial fraction as a sum of basic fractions, each with a polynomial of lower degree in the denominator. By handling these simpler fractions, integration becomes straightforward.

Understanding polynomial fractions and how to manipulate them is essential. It helps unlock the potential of more advanced integration techniques, allowing a clearer path to the solutions of calculus problems.