Problem 11 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the integrals. $$ \int \frac{3 x^{2}+3}{\left(x^{2}-1\right)(x-2)} d x $$

Step-by-Step Solution

Answer

The integral of a given fraction after further simplification and integration is a combination of different terms that include natural log \(\ln\) functions and a degree term, accompanied by the constant of integration, C.

Step 1: Partial Fraction Decomposition

The given integral can be decomposed into simpler fractions in the form \[\frac{ax+b}{x^2-1}\] and \[\frac{c}{x-2}\]. Now set \[\frac{3x^{2}+3}{(x^{2}-1)(x-2)}=\frac{ax+b}{x^2-1} + \frac{c}{x-2}\]. Multiply through by \[(x^{2}-1)(x-2)\] to clear the denominators. By comparing coefficients, find the values of a, b, and c.

Step 2: Integration

Having broken down the fraction, integrate the two parts separately. The integral of a form \[\frac{ax+b}{x^2-1}\] can be solved by using power rule for integration and \[\frac{1}{a} \ln|ax^2-1|\]. The integral of the form \[\frac{c}{x-2}\] can be solved as \[c\ln|x-2|\]. Apply these rules and integrate.

Step 3: Combine Results & Add Constant

Combine the results of the two integrations and add the constant of integration (C) to represent the indefinite integral.

Key Concepts

Partial Fraction Decomposition

When faced with complex rational expressions, breaking them down into simpler parts can be the key to solving integration problems. This is where partial fraction decomposition comes into play. Imagine trying to eat a large pizza; it’s much easier if you cut it into slices first! This process involves expressing a complex fraction as a sum of simpler fractions known as partial fractions. In our problem, \[ \frac{3x^{2}+3}{(x^{2}-1)(x-2)} \]is expressed as \[\frac{ax+b}{x^2-1} + \frac{c}{x-2} \]. To achieve this, you must first multiply through by the common denominator \((x^{2}-1)(x-2)\) to clear the fractions and compare coefficients. By equating the numerators on both sides of the equation and solving for each coefficient \(a, b,\) and \(c\), you effectively "slice" the fraction into smaller, more manageable pieces. This sets the stage for straightforward integration.

Power Rule for Integration

The power rule for integration is a fundamental tool, especially when dealing with polynomial expressions. If you’re familiar with the power rule from differentiation, where you bring down the power as a coefficient and reduce the exponent by one, integration does the opposite. For the integral of a term\(x^{n}\),where \(neq -1\), you add one to the exponent and divide by this new exponent. Formally, it's written as:

• \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\)

Applying this rule during the integration of partial fractions simplifies polynomial pieces. For example, in a decomposed term like \(\frac{ax+b}{x^2-1}\), you can break apart the numerator and apply the power rule to each separate term, making it possible to integrate effortlessly.

Logarithmic Integration

Logarithmic integration is employed when you recognize a derivative that closely resembles a fraction. Generally, when the integral has a form of \(\frac{1}{u} du\), it calls for logarithmic integration, yielding the result as

• \(\ln|u| + C\)

In our exercise, after decomposing the fraction into simpler terms, we encounter terms that fit this format. For instance, for the fraction\(\frac{c}{x-2}\),the integral is directly proportional to the logarithm:

• \(c \ln|x-2| + C\)

Recognizing such patterns is key to efficient integration, allowing you to convert a daunting task into a straightforward one step by step. This approach not only simplifies the computation but enhances your understanding of how logarithms interplay with integration.