Problem 48 - Applied Calculus

Question

Find each integral. [Hint: Try some algebra.] $$ \int(x+4)(x-2) d x $$

Step-by-Step Solution

Answer

The integral of \((x+4)(x-2)\) is \(\frac{x^3}{3} + x^2 - 8x + C\).

Step 1: Expand the Expression

To simplify the integral, let's first expand the expression inside the integral. This means apply the distributive property (also known as FOIL for binomials) to \((x+4)(x-2)\). Calculate the product:\[(x+4)(x-2) = x^2 - 2x + 4x - 8\]Combine like terms:\[x^2 + 2x - 8\]

Step 2: Rewrite the Integral

Replace the original expression in the integral with the expanded form:\[\int (x^2 + 2x - 8) \, dx\]

Step 3: Integrate Term by Term

Integrate each term separately using the power rule for integration. The power rule states that \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\):1. \(\int x^2 \, dx = \frac{x^3}{3} + C_1\)2. \(\int 2x \, dx = x^2 + C_2\)3. \(\int -8 \, dx = -8x + C_3\)Combine these results:\[\frac{x^3}{3} + x^2 - 8x + C\]

Step 4: Write the Final Answer

Combine all parts together for the final integrated expression:\[\int(x+4)(x-2) \, dx = \frac{x^3}{3} + x^2 - 8x + C\]

Key Concepts

Polynomial Integration

Polynomial integration is the process of finding the integral of polynomial expressions. Polynomials are composed of terms that include variables raised to non-negative integer powers.
Before integrating, it is beneficial to simplify or expand the expression if needed. This approach makes it easier to integrate the terms individually.

In the given exercise, we first expand

• (x+4)(x-2) = x^2 + 2x - 8

by using the distributive property. Expanding the polynomial helps us rewrite the integral in a form amenable to simple integration techniques. Once expanded, each term of the polynomial expression is individually integrated.

Power Rule in Calculus

The power rule is a fundamental technique in calculus for integrating expressions with variable terms. The power rule states:

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

where \(n\) is a real number and \(C\) represents the constant of integration.

Using the power rule in this exercise involves integrating each term of the expanded polynomial separately. For example:

• \( \int x^2 \, dx = \frac{x^3}{3} + C_1 \)
• \( \int 2x \, dx = x^2 + C_2 \)
• \( \int -8 \, dx = -8x + C_3 \)

We then combine these results to find the complete integral of the expression.

Algebraic Expansion

Algebraic expansion is a method used to simplify expressions, particularly those involving binomials. The technique involves applying the distributive property to multiply two binomial expressions.
In the context of this exercise, the expression \((x+4)(x-2)\) is expanded to:

• \( x^2 + 2x - 8 \)

This step is crucial because it translates the expression into a simpler form, where polynomial integration techniques, such as the power rule, can easily be applied. Using expansion, we break down the expression into distinct polynomial terms, which simplifies subsequent calculations and allows integrals to be more straightforwardly evaluated.