Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Use the Log Rule to find the indefinite integral. $$ \int \frac{x^{2}+2 x+3}{x^{3}+3 x^{2}+9 x+1} d x $$

Step-by-Step Solution

Answer

The indefinite integral of the given function is \(ln |x^{3}+3x^{2}+9x+1| + C.

Step 1: Identify the function and its derivative

In order to apply the log rule, we need to identify a function and its derivative in the integrand. In the given problem, consider the function \(u=x^{3}+3 x^{2}+9 x+1\). Then its derivative is \(du=(3x^{2} +6x +9)dx\).

Step 2: Simplify the integrand

The integrand can be rewritten as \[ \int \frac{x^{2}+2 x+3}{x^{3}+3 x^{2}+9 x+1} dx = \int \frac{1}{u} du\]. This involves distributing the dx and combining like terms in the numerator.

Step 3: Apply the Log Rule

Now, we can apply the log rule. The integral of \(1/u\) with respect to \(u\) is equal to \(ln |u|\), therefore the integral of our function will be \(ln |x^{3}+3x^{2}+9x+1|\) plus the constant of integration C.

Key Concepts

Logarithmic Integration

Logarithmic integration is a technique used to solve integrals that fit the form \(\int \frac{1}{u} \, du\). This type of integral often appears in problems where the derivative of the denominator is present in the numerator. It utilizes the natural logarithmic function for simplifying the integration process. To apply the log rule, identify a function \(u\) within the integrand and its derivative \(du\). Here, \(u\) should be a part of the denominator.
In our example, the function within the denominator is \(x^{3}+3x^{2}+9x+1\) and its derivative \(3x^{2}+6x+9\) is closely related to the numerator. This allows us to rewrite the integrand using \(u\) and \(du\).

• Make sure that the numerator is proportional to the derivative of the denominator before applying the logarithmic integration.
• Once the function is rewritten as \(\int \frac{1}{u} \, du\), you can successfully use the log rule.

The outcome of this integration step is the natural logarithm \(\ln|u| + C\), where \(C\) is the constant of integration.

Integration Techniques

There are several techniques to consider while solving indefinite integrals. Each technique provides a unique methodology tailored to different functional forms encountered in calculus.
One common technique involves substitution, where we replace a part of the integrand with a new variable to simplify the expression. In the given exercise, recognizing the derivative of one part involved consideration of substitution.
Another technique is partial fraction decomposition, often used when dealing with rational expressions where the degree of the numerator is less than the denominator after simplification.

• Logarithmic integration is especially useful when the integrand includes a fraction where the derivative of the denominator can be found easily in the numerator.
• Understanding when to use each technique is crucial for efficient problem-solving in integral calculus.

Selecting the appropriate technique is pivotal and often begins with identifying the type of function you are integrating.

Derivative Identification

In order to use many integration techniques effectively, derivative identification plays a key role. By recognizing derivatives in a problem, especially in integrands, you can simplify equations quickly.
The original exercise required identifying the function \(x^{3}+3x^{2}+9x+1\) and its derivative \(3x^{2}+6x+9\). Recognizing this pattern allows us to apply logarithmic integration to solve the integral.
Here are steps that illustrate how to identify the derivatives within integrands:

• Look for polynomial, trigonometric, exponential, and logarithmic functions alongside their derivatives.
• The derivative of a sum or difference is the sum or difference of the derivatives, so watch for this pattern in the given problem.
• Understand that multiplication by a constant factor does not affect the derivative identification process.

The ability to spot these derivatives quickly is essential for mastering various integration techniques and can greatly enhance your problem-solving speed and accuracy.