Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \int\left(x^{2}+x\right) d x $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Type of Integral

The given integral is \( \int (x^2 + x) \, dx \). This is a polynomial integral, which can be solved using basic integration rules for polynomials.

Step 2: Apply the Power Rule

The power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for any real number \( n eq -1 \). We will apply this rule to each term in the polynomial separately.

Step 3: Integrate Each Term Separately

Firstly, integrate \( x^2 \): \( \int x^2 \, dx = \frac{x^{3}}{3} \). Secondly, integrate \( x \): \( \int x \, dx = \frac{x^{2}}{2} \).

Step 4: Combine the Results

Combine the integrals of each term: \[ \int (x^2 + x) \, dx = \frac{x^3}{3} + \frac{x^2}{2} + C \] where \( C \) is the integration constant.

Key Concepts

Integration of Polynomials

When dealing with polynomial integration, we are interested in finding the "antiderivative" of a polynomial function. This means reversing the operation of differentiation. In our example, \( \int (x^2 + x) \, dx \), we are focusing on the polynomial \( x^2 + x \). Polynomial functions are expressions consisting of variables (like \( x \)) raised to various powers and multiplied by coefficients.

To integrate a polynomial:

• Break it down into separate terms.
• Apply integration rules to each term individually.

Polynomials are often straightforward to integrate because they follow consistent rules for each component. Remember, each term in a polynomial is integrated separately, making them easier to manage.

Power Rule for Integration

The power rule for integration is one of the most fundamental tools when working with polynomials. It enables you to integrate terms of the form \( x^n \). The rule is expressed as \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n \) is a real number and cannot equal \(-1\).

How to Apply the Power Rule

• Identify the exponent \( n \) of each term separately.
• Use the formula to find the antiderivative of each term.
• Add an integration constant \( C \) to account for all possible antiderivatives.

This rule transforms the task of integration into a simple algebraic process, making it highly user-friendly. It shows that if you can increase the exponent by one, then divide by this new exponent for each term, integration becomes a much more manageable task!

Indefinite Integrals

Indefinite integrals are integrals that do not have specified limits. In essence, they represent a family of functions based on the antiderivative of the integrand. When you solve an indefinite integral, like \( \int (x^2 + x) \, dx \), you find a general formula that represents all possible accumulated quantities of that function.

What Makes Them "Indefinite"

• They do not specify the limits of integration (upper and lower limits).
• The result includes a constant \( C \), representing a "family" of functions that all derivatives would map back to.

This concept highlights the fluid nature of integration: without a definite boundary, any constant could be added to the function, explaining the presence of \( +C \) in the solution. Therefore, indefinite integrals are a vital part of building understanding in calculus, offering insight into both processes and results.