Problem 51 - College Algebra and Calculus: An Applied Approach

Question

Writing What is the first step when integrating \(\int \frac{x^{2}}{x-5} d x ?\) Explain. (Do not integrate.)

Step-by-Step Solution

Answer

The first step in integrating \(\int \frac{x^{2}}{x-5} d x\) is to perform polynomial division to simplify the integral into \(\int \frac{(x-5)*x + 5x}{x-5} dx \).

Step 1: Identify the Integral

We need to evaluate the integral: \(\int \frac{x^{2}}{x-5} d x ?\).

Step 2: Determine the Integration Method

We analyze the integrand to determine the appropriate technique (direct integration, substitution, integration by parts, partial fractions, etc.).

Step 3: Perform the Integration

Applying the chosen method, we evaluate the integral step by step.

Step 4: State the Result

The first step in integrating \(\int \frac{x^{2}}{x-5} d x\) is to perform polynomial division to simplify the integral into \(\int \frac{(x-5)*x + 5x}{x-5} dx \).

Key Concepts

Polynomial Division

Polynomial Division is an algebraic technique used to simplify complex rational functions. When integrating a rational function like \( \int \frac{x^{2}}{x-5} dx \), where the numerator's degree is higher or equal to the denominator's, dividing the numerator by the denominator simplifies the problem. Just like in long division with numbers, you find how many times the denominator can 'fit' into the numerator. This process transforms the original integrand into a simpler expression, often resulting in a polynomial plus a fraction with a lower degree numerator.

To break this down further, imagine we're dividing \( x^2 \) by \( x-5 \). Initially, it might seem \( x-5 \) does not fit into \( x^2 \), but by reorganizing \( x^2 \) into \( (x-5)\cdot x + 5x \), we're effectively 'fitting' \( x-5\) into \( x^2 \) one time, plus an additional \( 5x \). The polynomial division simplifies the integrand to a form that can be easily integrated using basic integration rules. It's essential to become comfortable with polynomial division as it's a common first step in integrating complex rational functions.

Integral Calculus

Integral Calculus is a branch of mathematics focused on finding the antiderivatives of functions, which is the reverse process of differentiation. Integrating rational functions, in particular, can be challenging. However, many integration techniques, like polynomial division, can simplify these functions into a form where standard integration rules apply.

In the example \( \int \frac{x^{2}}{x-5} dx \), we see that before integrating, it's crucial to re-write the integrand into a simpler form using polynomial division. Only after simplifying should we proceed with finding the integral. Once simplified, the integral may often be computed by recognizing familiar forms or functions, such as polynomials, for which direct integration formulas exist. In this way, integral calculus not only involves the act of integrating itself but also preparatory steps like simplification to make the integration process more straightforward.

Algebraic Manipulation

Algebraic Manipulation encompasses various techniques used to reformat and simplify mathematical expressions. To successfully integrate a function like \( \int \frac{x^{2}}{x-5} dx \), one must often manipulate it into a more manageable form. This might involve factoring, expanding, or, as highlighted in our example, performing polynomial division.

The manipulation of \( x^2 \) into \( (x-5)*x + 5x \) to simplify the original integral is an example of algebraic manipulation. Through this process, we take an initial expression that is not straightforward to integrate and rework it into a format that is much easier to handle. Algebraic manipulation skills are vital for any student studying calculus, as they pave the way for making complex integrals solvable. Understanding these skills can also help in recognizing patterns and function behaviors that are fundamental to mastering calculus.