Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{x^{2}+19 x+10}{2 x^{4}+5 x^{3}} d x\)

Step-by-Step Solution

Answer

Decompose into partial fractions and integrate each term separately.

Step 1: Factor the Denominator

The first step in solving the integral using partial fraction decomposition is to factor the denominator of the integrand. The denominator is \(2x^4 + 5x^3\). We can factor out the greatest common factor, which in this case is \(x^3\), leading to \(x^3(2x + 5)\).

Step 2: Set Up the Partial Fraction Decomposition

Now express the integrand \(\frac{x^2 + 19x + 10}{x^3(2x + 5)}\) as a sum of simpler fractions. These fractions will have the form: \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{2x + 5}\).

Step 3: Write the Equation for Decomposition

Combine the right-hand side to match the original fraction: \(\frac{A(2x + 5) + Bx(2x + 5) + Cx^2(2x + 5) + Dx^3}{x^3(2x + 5)}\), which should equal \(x^2 + 19x + 10\).

Step 4: Solve for Coefficients

Expand and equate the numerators: \(A(2x + 5) + Bx(2x + 5) + Cx^2(2x + 5) + Dx^3 = x^2 + 19x + 10\). Solve this equation to find the values of \(A\), \(B\), \(C\), and \(D\).

Step 5: Integrate Each Term Separately

Once you have the values of \(A\), \(B\), \(C\), and \(D\), integrate each fraction separately: - \(\int \frac{A}{x} \, dx = A \ln|x| + C_1\)- \(\int \frac{B}{x^2} \, dx = -\frac{B}{x} + C_2\)- \(\int \frac{C}{x^3} \, dx = -\frac{C}{2x^2} + C_3\)- \(\int \frac{D}{2x + 5} \, dx = \frac{D}{2} \ln|2x + 5| + C_4\).

Step 6: Combine Integrated Terms

Combine all the integrated terms: \[ A \ln|x| - \frac{B}{x} - \frac{C}{2x^2} + \frac{D}{2} \ln|2x + 5| + C \] where \(C\) is the constant of integration.

Key Concepts

Integration Techniques

Integration techniques are essential tools in calculus that help us find the antiderivatives of functions, which are often crucial in solving calculus problems such as evaluating definite and indefinite integrals. One particularly useful technique is partial fraction decomposition. This technique involves breaking down complex rational expressions into simpler fractions, making them easier to integrate. For the integral \(\int \frac{x^{2}+19 x+10}{2 x^{4}+5 x^{3}} \, dx\), partial fraction decomposition is used to simplify the integration process. By expressing a complex rational expression as a sum of simpler fractions, each term can be integrated separately. This method is especially helpful when the original fraction has a complicated polynomial in the denominator, as it converts the problem into a series of straightforward integrations like \(\int \frac{A}{x} \, dx\) and others.

Calculus Problem Solving

Solving calculus problems often requires breaking down a complex problem into simpler parts, a strategy that is highly effective with integration. In the exercise at hand, the goal was to integrate a rational function using partial fraction decomposition. To do this, we need to use several problem-solving steps:

• First, recognize that partial fraction decomposition is applicable because the degree of the numerator is less than that of the denominator.
• Then, factor the denominator, which allows us to determine the appropriate form of partial fractions needed.
• Set up an equation from the fraction decomposition to find the coefficients that simplify the expression.
• Finally, integrate each decomposed term separately, using basic integral forms.

By systematically applying these steps, students can tackle even intimidating calculus problems with confidence and achieve the correct solution.

Factor the Denominator

Factoring the denominator is a crucial step in partial fraction decomposition, as it lays the foundation for expressing the integrand as a sum of simpler fractions. In our given problem, the denominator is \(2x^4 + 5x^3\). The first step in factoring is to find the greatest common factor (GCF) of the terms, which in this case is \(x^3\). Factoring out \(x^3\) from the denominator gives us \(x^3(2x + 5)\).

Why is this important? By factoring the denominator, we can identify the different fractions in our decomposition:

• Simple fractions characterized by powers of \(x\) such as \(\frac{A}{x}\), \(\frac{B}{x^2}\), and \(\frac{C}{x^3}\).
• Fractions involving linear terms like \(\frac{D}{2x + 5}\).

This breakdown transforms a complex integration problem into several smaller and manageable tasks. Each term corresponds to a specific part of the factored denominator, allowing for an organized approach to finding unknown coefficients and eventually solving the integral.