Problem 47 - Applied Calculus

Question

47-50. Find each integral. [Hint: Try some algebra.] $$ \int(x+1) x^{2} d x $$

Step-by-Step Solution

Answer

\( \int (x+1) x^2 \, dx = \frac{x^4}{4} + \frac{x^3}{3} + C \)

Step 1: Expand the Polynomial

First, expand the expression \( (x+1) \times x^2 \) using distributive property. This results in: \[ x^3 + x^2 \].

Step 2: Integrate Each Term Separately

Now that you have the expression expanded to \( x^3 + x^2 \), integrate each term separately. The integral of \( x^3 \) is: \[ \int x^3 \, dx = \frac{x^4}{4} + C_1 \] The integral of \( x^2 \) is:\[ \int x^2 \, dx = \frac{x^3}{3} + C_2 \].

Step 3: Combine the Integrals

Combine the results of the individual integrals, using a single constant of integration \( C \):\[ \int (x^3 + x^2) \, dx = \frac{x^4}{4} + \frac{x^3}{3} + C \].

Key Concepts

Algebraic Expansion

Algebraic expansion is a crucial step in simplifying expressions, especially when dealing with integrals. It involves converting complex expressions into simpler terms that are easier to work with. In our exercise, the expression \(x+1\) is multiplied by \ x^2\. We use the distributive property to expand the expression. This means we distribute \(x+1\) across \ x^2 \, which results in the expanded form \(x^3 + x^2\). By breaking it down, we gain an expression that's simpler to integrate. Essentially, algebraic expansion makes mathematical operations, like integration, more straightforward. Remember: always look for opportunities to expand expressions before integrating. It reduces complexity and clarifies the process.

Definite and Indefinite Integrals

Integrals are a fundamental part of calculus, divided into two main types: definite and indefinite. In our exercise, we're dealing with indefinite integrals, as indicated by the arbitrary constant \(C\).

• **Indefinite Integrals:** These do not have specified upper and lower limits. Instead, they represent a family of functions and include a constant of integration, \(C\), to account for all possible values of an antiderivative.

In contrast:

• **Definite Integrals:** These have specific upper and lower limits. They calculate the net area under the curve of a function over a certain interval. There's no constant involved as the limits provide exact values.

Understanding the distinction is vital. In practical applications, definite integrals solve problems like finding areas, while indefinite integrals find general solutions, characterizing a family of possible functions.

Polynomial Integration

Polynomial integration is one of the more straightforward techniques within integral calculus, due to the predictable nature of polynomials. In this process, each term in the polynomial is integrated separately.For example, in the solution:

• The term \(x^3\) is integrated to \(\frac{x^4}{4}\ + C_1\).
• The term \(x^2\) becomes \(\frac{x^3}{3}\ + C_2\).

The rule for integrating a polynomial term is straightforward: for any term \(ax^n\), its integral is \(\frac{ax^{n+1}}{n+1} \), where \(a\) is a constant and \(n\) is a whole number.Polynomial integration simplifies the integral of expanded algebraic expressions. Mastering this helps tackle a wide variety of problems efficiently by providing an easy way to arrive at antiderivatives. Once each term is integrated, combine them, paying attention to integration constants for a complete solution.