Problem 6 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(5-14,\) evaluate the integral. $$\int \frac{2 x+16}{x^{2}+x-6} d x$$

Step-by-Step Solution

Answer

The integral of the given function is \(\log|x^{2}+x-6| + 15x + C\).

Step 1: Identify the derivative of the denominator

On examining the function, the denominator is \(x^{2}+x-6\). Taking the derivative of this expression, we obtain \(2x + 1\). The numerator is \(2x + 16\), which isn't exactly the derivative of the denominator but if we rewrite \((2x+16)\) as \((2x+1+15)\), we can see that \(2x+1\) is the derivative of \(x^{2}+x-6\) and \(15\) is an additional constant.

Step 2: Apply the rule of integration

The function under the integral is now written in the form of \(f'(x)/f(x) + constant\). Considering only the \(\int (2x+1)/(x^{2}+x-6) dx\) part of the integral, by using the rule of integration, \(\int f'(x)/f(x) dx = \log|f(x)|\), we get \(\log|x^{2}+x-6|\).

Step 3: Integrate the constant

Since \(\int du = u + C\), the integral of the constant 15 with respect to x would simply be \(15x\).

Step 4: Combine the results and add the integration constant C

Since the original problem is the integral of a sum of two terms, the integral of the whole is just the sum of the integrals of the elements. Combining results from Step 2 and Step 3 and adding the constant of integration \(C\), the integral of the given function is \(\log|x^{2}+x-6| + 15x + C\).

Key Concepts

Integration Techniques

Integration techniques are mathematical methods used to find the integral of a function. In this exercise, we deal with splitting the integral into simpler parts. The problem involves a rational function, which often warrants decomposing the expression for easier integration. One useful approach is to express the numerator in terms of the derivative of the denominator plus an extra constant.

• Recognize the structure: This helps in identifying parts of the function that align nicely with known integration rules.
• Adjust the function: Rewrite the numerator to match the derivative of the denominator, making the problem simpler to handle.
• Break down the integral: Separate the modified integral into manageable parts for straightforward computation.

Using these steps, we simplify the integration problem and make it more approachable.

Derivative

A derivative represents the rate at which a function changes at any given point. In the context of integration, identifying the derivative of the denominator is the key to simplifying the structure of the integral. To solve the original exercise, we need to understand the denominator's role:

• Function of the denominator: Given as \( x^2 + x - 6 \).
• Derivative process: Apply basic rules to find that the derivative is \( 2x + 1 \).

This information informs how we rewrite parts of the numerator, facilitating a more straightforward integration using known formulas.

Logarithmic Integration

Logarithmic integration utilizes the natural logarithm function to solve integrals where a fraction of a derivative over its original function appears.This happens when:

• You have a form \( \frac{f'(x)}{f(x)} \) within the integral.
• The corresponding rule \( \int \frac{f'(x)}{f(x)} \, dx = \log|f(x)| + C \) applies directly.

In our exercise, after recognizing that \( 2x+1 \) is the derivative of the denominator \( x^2+x-6 \), we use logarithmic integration. This results in part of our solution being \( \log|x^{2}+x-6| \), capturing the essence of the log integration technique.

Constant of Integration

The constant of integration, denoted as \( C \), is a critical component when solving indefinite integrals. It represents an unknown constant that can take any value since integration results in a family of functions. For our exercise:

• When integrating, always add \( C \) to accommodate all potential functions that could differentiate back to the original integrand.
• It emphasizes that indefinitely integrated results aren't unique until further conditions or information about the function are provided.

Therefore, after computing the integral \( \log|x^{2}+x-6| + 15x \), we add \( C \), ensuring all possible scenarios of antiderivatives are covered.