Problem 18 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the integrals. $$ \int \frac{2 x^{3}-2 x^{2}+4 x+8}{(x-2)^{2}\left(x^{2}+3\right)} d x $$

Step-by-Step Solution

Answer

\[3 ln|x-2| - \frac{2}{x-2} + \frac{1}{2} ln|x^2+3| - arctan(\frac{x}{\sqrt{3}}) + C\]

Step 1: Decompose into Partial Fractions

We need to decompose the fraction into simpler fractions. This can often be accomplished by using long division or simple comparison of coefficients. Writing the function \( \frac{2 x^{3}-2 x^{2}+4 x+8}{(x-2)^{2}(x^{2}+3)} \) in the general form \( \frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{Cx+D}{x^2+3} \), equating like terms, and then solving the ensuing system of linear equations gives us the values as A = 3, B = 2, C=1 and D=1.

Step 2: Integrate Each Fraction Separately

After decomposing into partial fractions, we will have \( 3 \int \frac{1}{x-2} dx + 2 \int \frac{1}{(x-2)^2} dx + \int \frac{x+1}{x^2 + 3} dx \) . Now we can integrate each of these separately using standard integration rules. The first integral can be solved by applying the natural logarithm rule, the second can be solved by power rule, and the last integral can be solved by substitution (let u = \( x^2 + 3 \) ).

Step 3: Combine the Results

We can find the integral of the given function by adding the integrals obtained in the previous step. They are \(3 ln|x-2| - \frac{2}{x-2} + \frac{1}{2} ln|x^2+3| - arctan(\frac{x}{\sqrt{3}}) + C\), where C is the constant of integration.

Key Concepts

Integration Techniques

To solve complex integrals involving rational functions, we often employ different integration techniques. One popular technique is partial fraction decomposition. This method is useful when dealing with polynomial fractions, allowing us to break down a complex expression into simpler, more manageable fractions.

Once decomposed, each simpler fraction can be integrated separately using known formulas, such as:

• The natural logarithm rule: \( \int \frac{1}{x-a} \, dx = \ln|x-a| + C \)
• The power rule: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \)
• Substitution, where we let a part of the integral become a new variable, often simplifying the integration process.

By leveraging these techniques, we transform a complicated integral into several simpler ones, making the problem more approachable.

Integral Calculus

Integral calculus is the branch of mathematics concerned with integrals and their applications. It allows us to calculate areas, volumes, and solve various physical problems. In the context of rational functions, integral calculus helps us find the antiderivative of functions expressed as a quotient of polynomials.

The process usually involves:

• Identifying a suitable method to simplify the integral, such as partial fraction decomposition or substitution.
• Applying standard integration formulas to the simplified terms.
• Combining the results, along with the constant of integration \( C \), to get the complete solution.

Through meticulous application of these steps, integral calculus enables us to solve complex integrals systematically and effectively.

Rational Functions

Rational functions are expressions represented as the ratio of two polynomials. They are expressed generally as \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. In integration, rational functions often require special methods to effectively evaluate their integrals.

A common approach to integrate rational functions includes:

• Performing polynomial long division if the degree of the polynomial in the numerator is greater than or equal to the degree of the polynomial in the denominator.
• Using partial fraction decomposition to break down the rational function into simpler fractions that are easier to integrate.

These techniques allow us to handle integrals involving rational functions by reducing them into a sum of standard integrals, making the process more straightforward. Understanding rational functions and their properties is essential for tackling integration problems in calculus.