Integration: Antiderivatives and Indefinite Integrals

Chapter 5: Integration

Antiderivatives and Indefinite Integrals

Integration is a fundamental concept in calculus, closely related to differentiation. The process of finding an antiderivative, or indefinite integral, allows us to recover a function from its derivative. This section introduces the concept of antiderivatives, their properties, and applications in business contexts.

Antiderivatives

• Definition: A function F is an antiderivative of a function f if $F'(x) = f(x)$.

• Family of Functions: Antidifferentiation does not yield a unique function, but an entire family of functions differing by a constant.

• Theorem: If the derivatives of two functions are equal on an open interval $(a, b)$, then the functions differ by at most a constant. Symbolically, if $F'(x) = G'(x)$ for all $x$ in $(a, b)$, then $F(x) = G(x) + k$ for some constant $k$.

Example: A Family of Antiderivatives

• Problem: Find all antiderivatives of $f(x) = x$.

• Solution: The general antiderivative is $F(x) = \frac{1}{2}x^2 + C$, where $C$ is any constant.

• Graphical Interpretation: The graphs of these antiderivatives are vertical translations of each other, passing through points such as (0,0), (0,1), and (0,2).

• Relationship: As shown in the graphs, the three antiderivatives are vertical translations of each other.

Indefinite Integrals: Formulas and Properties

• Notation: The indefinite integral of $f(x)$ with respect to $x$ is written as $\int f(x)\,dx$.

• Constant of Integration: The arbitrary constant $C$ is called the constant of integration.

• Reverse Operations: Indefinite integration and differentiation are reverse operations, except for the addition of the constant of integration. Symbolically, $\frac{d}{dx}\left(\int f(x)\,dx\right) = f(x)$ and $\int f'(x)\,dx = f(x) + C$.

Formulas: Indefinite Integrals of Basic Functions

• Power Rule: $\int x^n\,dx = \frac{1}{n+1}x^{n+1} + C$, for $n \neq -1$

• Exponential Rule: $\int e^{x}\,dx = e^{x} + C$

• Constant Rule: $\int a\,dx = a x + C$

• Sum Rule: $\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx$

Properties: Indefinite Integrals

• Linearity: $\int [a f(x) + b g(x)]\,dx = a \int f(x)\,dx + b \int g(x)\,dx$

• Integration is the reverse of differentiation: $\int f'(x)\,dx = f(x) + C$

Examples Using Indefinite Integral Properties and Formulas

• Example: Integrate $f(x) = 3x^2 + 2x + 5$.

• Solution: $\int (3x^2 + 2x + 5)\,dx = 3 \int x^2\,dx + 2 \int x\,dx + 5 \int dx = x^3 + x^2 + 5x + C$

Applications of Antiderivatives

• Particular Antiderivative: Find the equation of the curve that passes through (3, 5) if the slope of the curve is given by $\frac{dy}{dx} = 2x$ at any point $x$.

• Solution: Integrate $2x$ to get $y = x^2 + C$. Use the point (3,5) to solve for $C$: $5 = 9 + C \Rightarrow C = -4$. Thus, $y = x^2 - 4$.

• Cost Function Example: If the marginal cost is $C'(x) = 0.3x^2 + 2x$ with fixed costs of $3,000$, find $C(x)$ and the cost of producing 30 units.

• Solution: Integrate $C'(x)$: $C(x) = 0.1x^3 + x^2 + C$. Use $C(0) = 3,000$ to find $C = 3,000$. Thus, $C(x) = 0.1x^3 + x^2 + 3,000$. For $x = 30$, $C(30) = 0.1 \times 27,000 + 900 + 3,000 = 2,700 + 900 + 3,000 = 6,600$.

• Advertising Example: A radio station wants to increase listeners from 27,000 to 41,000. The campaign should last about 40 more days, based on the integration of the growth rate function.

Summary Table: Indefinite Integral Rules

Function	Indefinite Integral
$x^n$	$\frac{1}{n+1}x^{n+1} + C$
$e^x$	$e^x + C$
$a$	$a x + C$
$f(x) + g(x)$	$\int f(x)\,dx + \int g(x)\,dx$

Additional info: The examples and applications provided are typical in business calculus, illustrating how integration is used to solve real-world problems such as cost analysis and growth modeling.