Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=3 x^{2}+\sqrt{3} $$

Step-by-Step Solution

Answer

The general antiderivative is \( F(x) = x^3 + \sqrt{3}x + C \).

Step 1: Identify the terms of the function

The given function is \( f(x) = 3x^2 + \sqrt{3} \). This function is composed of two terms: \( 3x^2 \) and \( \sqrt{3} \).

Step 2: Apply the power rule to the first term

For the term \( 3x^2 \), we use the power rule of integration which states \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). Therefore, the antiderivative of \( 3x^2 \) is \( \frac{3x^{2+1}}{2+1} = x^3 \).

Step 3: Integrate the constant term

The antiderivative of a constant \( a \) is \( ax + C \). Therefore, the antiderivative of \( \sqrt{3} \) is \( \sqrt{3}x \).

Step 4: Combine the results

Combine the antiderivatives from Steps 2 and 3 to form the general antiderivative. The combined antiderivative is \( F(x) = x^3 + \sqrt{3}x + C \).

Key Concepts

Integration

Integration is a fundamental concept in calculus often introduced along with differentiation. In essence, integration is the reverse process of differentiation. It allows us to find a function given its derivative. The function obtained from integrating is called the antiderivative or integral.

• The integral of a function gives the accumulation of the quantities, which can be thought of as finding the area under a curve represented by the function.
• Definite integrals have limits of integration and compute a real value, while indefinite integrals do not have limits and provide a family of functions denoted by adding the constant of integration.

In our example, the task is to find the general antiderivative of the function given as a sum of terms. Different terms in a function can often be separately integrated and then summed. The process leverages formulas and rules developed for different types of functions, such as polynomials and constants, to simplify the integration process.

Power Rule

The Power Rule is a very useful tool in both differentiation and integration. In integration, it helps find antiderivatives of power functions, which are functions of the form \( x^n \), where \( n \) is any real number.

• The Power Rule for integration states: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{where } n eq -1. \]
• This means we simply add 1 to the power and then divide by the new power.

In the given exercise, this rule was applied to the term \( 3x^2 \) by taking \( n = 2 \). The antiderivative becomes \( \frac{3x^3}{3} = x^3 \). The rule simplifies the process, making it easy to handle polynomial functions.

Constant of Integration

In calculus, when we find an indefinite integral, we usually end up with a general solution that includes a constant term, known as the Constant of Integration. This is because the derivative of a constant is zero, which creates an infinite set of solutions that differ by a constant.

• The Constant of Integration is typically denoted by \( C \) and represents any real number.
• It ensures that the set of all possible original functions is captured, since adding any constant to a function doesn't change its derivative.

In our example, after integrating all terms of the function \( f(x) = 3x^2 + \sqrt{3} \), a \( C \) is added to form the final general antiderivative: \( F(x) = x^3 + \sqrt{3}x + C \). This expression includes the entire family of functions that could have \( f(x) \) as their derivative.

Mathematics Education

In mathematics education, understanding the process of integration and its rules is essential for students. Learning integration fosters a deeper comprehension of calculus and its applications.

• Students build upon foundational math skills such as algebra and differentiation when learning to integrate.
• Recognizing patterns and applying rules such as the Power Rule facilitate efficient problem solving.
• Guided practice with examples, like the one provided here, aids students in mastering the topic by seeing step-by-step procedures and explanations.

The integration problem described here highlights practical applications of theory taught in mathematics education. It shows how abstract concepts are used in real problem-solving scenarios, enhancing the learning experience and preparing students for advanced studies.