Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The given function is a sum of two terms: a power function and a constant. The function to integrate is \( f(x) = x^2 + \pi \). We will find the antiderivative of each term separately.

Step 2: Antiderivative of the Power Term

The antiderivative of \( x^n \) is given by \( \frac{x^{n+1}}{n+1} + C \). For the power term \( x^2 \), we use \( n = 2 \). Thus, its antiderivative is \( \frac{x^{2+1}}{2+1} = \frac{x^3}{3} \).

Step 3: Antiderivative of the Constant Term

The antiderivative of a constant \( c \) is \( cx + C \). For our constant \( \pi \), the antiderivative is \( \pi x \).

Step 4: Combine the Antiderivatives

Combine the antiderivatives of both terms to find the general antiderivative of the given function. This yields \( F(x) = \frac{x^3}{3} + \pi x + C \), where \( C \) is the constant of integration.

Key Concepts

Power Function

A power function is a mathematical expression of the form \( x^n \), where \( n \) is a real number known as the exponent. In the context of calculus, understanding how to find the antiderivative of a power function is crucial. When integrating a power function, you apply the power rule for antiderivatives. This rule states that the antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \), where \( C \) represents the constant of integration.
This rule helps in finding the area under the curve or in reversing differentiation to find the original function. For example, the function \( x^2 \) is a simple power function with \( n = 2 \). Applying the power rule, its antiderivative becomes \( \frac{x^{2+1}}{2+1} = \frac{x^3}{3} \). Understanding this fundamental rule is essential for solving more complex calculus problems.

Constant Function

A constant function is one where the output value is the same, no matter the input. In simpler terms, if you have a constant number like \( \pi \) in a function, it doesn't change, no matter the value of \( x \). Integrating a constant function is straightforward.
When integrating a constant \( c \), the antiderivative is expressed as \( cx + C \). Here, \( C \) is the integration constant that incorporates all constants determined by the boundary conditions. For example, the constant \( \pi \) in our function becomes \( \pi x \) when we find its antiderivative. This rule allows us to accumulate the constant along the x-axis as we integrate. Recognizing and applying these rules help make the integration process more manageable.

Integration

Integration is a fundamental concept in calculus that involves finding the antiderivative of a function. It is essentially the reverse process of differentiation. Through integration, you can calculate areas under curves, total accumulated quantities, or the original function from its derivative.

• The indefinite integral process determines the general antiderivative of a function, giving a family of functions distinguished by a constant of integration \( C \).
• Each integral calculation must consider power functions, constant functions, and their respective integration rules.

For the function \( f(x) = x^2 + \pi \), integration involves finding separate antiderivatives for each term: applying the power rule for \( x^2 \), and the constant integration rule for \( \pi \). Afterwards, these results are combined to form the general solution, influencing how calculations in physics and engineering are often performed.

Calculus

Calculus is a branch of mathematics focused on change and motion, through differentiation and integration. It encompasses methods for solving problems involving rates of change (differentiation) and accumulation (integration). These are two core pillars that often appear in physics, engineering, economics, and other fields.

• Integration, one of the main operations in calculus, helps determine total accumulation and reverse differential processes to retrieve the original functions.
• Differentiation, on the other hand, deals with finding rates of change and slopes of curves.

In solving problems like finding the antiderivative of \( f(x) = x^2 + \pi \), calculus principles such as using appropriate antiderivative rules are applied. These principles facilitate understanding complex systems, develop computational techniques, and help in solving real-world applications efficiently.