Problem 12 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int \frac{x+2}{x^{2}+4 x+2} d x\)

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{1}{2} \ln|x^2 + 4x + 2| + C \).

Step 1 - Simplify the Integrand

First, we will simplify the integrand \(\frac{x+2}{x^{2}+4x+2}\). Notice that the denominator can be factored or completed the square if needed, but here, we look for simpler approaches. Let’s split it into simpler fractions.

Step 2 - Use Partial Fraction Decomposition

Attempt to rewrite \(\frac{x+2}{x^2 + 4x + 2}\) as a linear combination of simpler fractions. Since the denominator is irreducible, we attempt a different substitution or recognition.

Step 3 - Use Substitution Method

Let \(u = x^2 + 4x + 2\). Then, the derivative \(du = (2x + 4) dx\) or \( \frac{du}{2} = (x+2)dx\). Now substitute into the integral: \(\frac{1}{2} \int \frac{1}{u} du\).

Step 4 - Integrate

Now the integral becomes \( \frac{1}{2} \int \frac{1}{u} du = \frac{1}{2} \ln|u| + C\).

Step 5 - Substitute Back

Replace \(u\) with the original expression: \( \frac{1}{2} \ln|x^2 + 4x + 2| + C\).

Key Concepts

partial fraction decomposition

Partial fraction decomposition is a technique to break down complex fractions into simpler parts that can be more easily integrated. This method is especially useful when dealing with rational functions, where the numerator and denominator are both polynomials. Here’s how it works:

• Identify the degree of the numerator and denominator. If the degree of the numerator is higher, perform polynomial division first.
• Factor the denominator into simpler polynomials, if possible.
• Express the original fraction as a sum of fractions with these simpler denominators.

This process simplifies the integral by allowing you to work with simpler fractions. However, in the given exercise, the denominator \(x^2 + 4x + 2\) is irreducible. Therefore, a different technique is required, leading us to the substitution method.

substitution method

The substitution method helps transform a complicated integral into a simpler form by changing variables. In our exercise, we use substitution to simplify the integrand. Here’s how the method works:

• Choose a substitution that simplifies the integral. In the given problem, we let \(u = x^2 + 4x + 2\).
• Differentiate the substitution to find \(du\) in terms of \(dx\). For our problem, \(du = (2x + 4)dx\).
• Rewrite the integral in terms of \(u\) and \(du\). Note that \(x+2 = \frac{du}{2}\), which simplifies our integral to \(\frac{1}{2} \int \frac{1}{u} du\).

This substitution transforms the original problem into one that involves basic integration techniques, making it easier to solve.

integration techniques

Integration techniques are various methods used to find the antiderivatives or indefinite integrals of functions. Common techniques include:

• Basic Integration: Use fundamental formulas like \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\).
• Substitution Method: Simplifies the integral by changing variables. Used effectively in our exercise.
• Partial Fraction Decomposition: Breaks a complex fraction into simpler fractions. Applicable when the denominator factorizes easily.
• Integration by Parts: Used when the integral involves a product of functions. Based on the formula \(\int u dv = uv - \int v du\).

In the exercise, partial fraction decomposition was initially considered but was not applicable due to the irreducible quadratic denominator. Instead, substitution method effectively converted the integral into a form easily integrated using basic techniques. The final result, after integrating and substituting back, is \(\frac{1}{2} \ln|x^2 + 4x + 2| + C\).