Problem 45 - College Algebra and Calculus: An Applied Approach

Question

(a) perform the integration in two ways: once using the Simple Power Rule and once using the General Power Rule. (b) Explain the difference in the results. (c) Which method do you prefer? Explain your reasoning. $$ \int x\left(x^{2}-1\right)^{2} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{1}{6}x^6 - \frac{1}{2}x^4 + \frac{1}{2}x^2 + C \). Both the Simple Power Rule and the General Power Rule give the same result. The choice of method depends on the specific situation and personal preferences.

Step 1: Integration using the Simple Power Rule

Let's start by multiplying out \( x\left(x^{2}-1\right)^{2} \) to express the integrand as a sum of terms with powers of \( x \). This gives us the polynomial \( x^5 - 2x^3 + x \). We'll use the Simple Power Rule, which says that the integral of \( x^n \) is \( \frac{1}{n+1}x^{n+1} \), we then get: \[ \int x\left(x^{2}-1\right)^{2} d x = \int (x^5 - 2x^3 + x) dx = \frac{1}{6}x^6 - \frac{1}{2}x^4 + \frac{1}{2}x^2 + C \] where \( C \) represents the constant of integration.

Step 2: Integration using the General Power Rule

The General Power Rule is a technique of integration that involves substitution and is usually useful when the integrand is a function of a polynomial in the variable of integration. We'll set \( u = x^2 - 1 \), then \( du = 2xdx \). Expressing the integrand in terms of \( u \) gives \( \frac{1}{2}u^2du \). We then use the power rule to find the integral: \[ \int \frac{1}{2}u^2du = \frac{1}{6}u^3 + C = \frac{1}{6}(x^2-1)^3 + C \] After expansion, this expression simplifies to the same result as when using the Simple Power Rule.

Step 3: Comparing Results and Choosing a Preferred Method

Both methods gave the same result for the integral, as they should. The choice of which method to use often depends on the specific integrand and personal preference. The Simple Power Rule can be quicker to apply if the integrand is already a sum of terms with powers of \( x \), but requires algebraic simplification of the integrand in advance. On the other hand, the General Power Rule can be more direct for integrands that are functions of a polynomial in \( x \), because it avoids the need for this simplification.

Key Concepts

Simple Power Rule

The Simple Power Rule is a straightforward technique used to find the integral of powers of a variable. When you see a term like \(x^n\) within an integrand, the Simple Power Rule makes integration swift and easy. Here's how it works: if you have a term \(x^n\), its integral is given by \(\frac{1}{n+1}x^{n+1}\), where \(n\) is the exponent of \(x\). Don't forget the constant of integration, \(C\), which we always add at the end.
To illustrate, consider integrating \(x^5 - 2x^3 + x\). Applying the Simple Power Rule separately to each term, you get:

• Integrating \(x^5\): \(\frac{1}{6}x^6\)
• Integrating \(-2x^3\): \(-\frac{1}{2}x^4\)
• Integrating \(x\): \(\frac{1}{2}x^2\)

After combining these results, our integral becomes \(\frac{1}{6}x^6 - \frac{1}{2}x^4 + \frac{1}{2}x^2 + C\). This method is clear and relies on performing straightforward algebraic manipulation.

General Power Rule

The General Power Rule involves a bit more finesse than the Simple Power Rule and is particularly useful when dealing with functions of polynomials. Unlike the Simple Power Rule, this technique focuses on using substitution to simplify the integration process. It often saves time by reducing the complexity of the original expression.
The essence of the General Power Rule is to choose appropriate substitution that simplifies the integrand. In our example, we set \(u = x^2 - 1\), making the differential \(du = 2xdx\). Rewriting the integral in terms of \(u\), we had \(\frac{1}{2}u^2 du\).

• After integrating \(\frac{1}{2}u^2\), we find \(\frac{1}{6}u^3 + C\).
• Substitute back \(u = x^2 - 1\) to give \(\frac{1}{6}(x^2-1)^3 + C\).

The General Power Rule is handy when confronted with composite functions, allowing you to tackle the problem without multiplying out and simplifying extensive polynomial terms.

Polynomial Integration

Polynomial integration is a method that leverages both the versatility of basic power rules and the strategies of substitution techniques like the General Power Rule. As seen in our example, polynomials can be straightforwardly integrated term by term using the Simple Power Rule, or by applying appropriate substitutions as in the General Power Rule.
When integrating a polynomial:

• Ensure each term of the polynomial is separated and manageable.
• Decide if simple power strategies suffice, or if substitution can bring dividends in simplification.

Polynomial integration is inherently systematic. By looking at individual components like \(x^5, -2x^3,\) and \(x\), the process becomes highly organized. Alternatively, when polynomial terms suggest a repeated substructure (like \((x^2 - 1)^2\)), opting for the General Power Rule can streamline the integration process. This structure ensures that whether through detailed expansion or efficient substitution, the integral can be evaluated proficiently.