Section 7.1: Antiderivatives – Business Calculus Study Notes

Section 7.1: Antiderivatives

Introduction to Antiderivatives

An antiderivative of a function is a function whose derivative is the original function. Antidifferentiation is the reverse process of differentiation and is fundamental in solving problems involving accumulation, such as finding areas, total change, and inverting rates.

• Definition: If $F'(x) = f(x)$, then $F(x)$ is an antiderivative of $f(x)$.

• There are infinitely many antiderivatives for a given function, differing by a constant.

• The process of finding an antiderivative is called integration.

General Form of Antiderivatives

• The general antiderivative of $f(x)$ is $F(x) + C$, where $C$ is an arbitrary constant.

• This is because the derivative of a constant is zero, so adding any constant to an antiderivative yields another antiderivative.

Example: If $f(x) = 2x$, then $F(x) = x^2$ is an antiderivative, but so is $F(x) = x^2 + 5$.

Indefinite Integral Notation

• The indefinite integral of $f(x)$ is written as $\int f(x)\,dx = F(x) + C$.

• The symbol $\int$ denotes integration, $f(x)$ is the integrand, $dx$ indicates the variable of integration, and $C$ is the constant of integration.

Rules of Antiderivatives

Power Rule

• For any real number $n \neq -1$:

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$

• This rule is the reverse of the power rule for derivatives.

Constant Multiple Rule and Sum/Difference Rule

• For any real number $k$:

$\int k f(x)\,dx = k \int f(x)\,dx$

• For sums and differences:

$\int [f(x) \pm g(x)]\,dx = \int f(x)\,dx \pm \int g(x)\,dx$

• The antiderivative of a sum/difference is the sum/difference of the antiderivatives.

Antiderivatives of Exponential Functions

• For the exponential function $e^{ax}$:

$\int e^{ax}\,dx = \frac{1}{a}e^{ax} + C$

• For general base $a > 0$, $a \neq 1$:

$\int a^{x}\,dx = \frac{1}{\ln a} a^{x} + C$

Antiderivative of $1/x$

• The antiderivative of $1/x$ is the natural logarithm:

$\int \frac{1}{x}\,dx = \ln|x| + C$

Worked Examples

• Example 1: $\int (3x^2 - 5x + 6)\,dx = x^3 - \frac{5}{2}x^2 + 6x + C$

• Example 2: $\int (2e^{3x} - 4)\,dx = \frac{2}{3}e^{3x} - 4x + C$

• Example 3: $\int \frac{1}{x}\,dx = \ln|x| + C$

Applications in Business Calculus

• Antiderivatives are used to find total cost, revenue, or profit functions from marginal functions.

• They are also used to solve problems involving accumulation, such as total change over time.

Example: If the marginal cost $C'(x) = 2x + 3$, then the total cost function is $C(x) = x^2 + 3x + C$.

Summary Table: Common Antiderivatives

Function $f(x)$	Antiderivative $F(x)$
$x^n$ ($n \neq -1$)	$\frac{x^{n+1}}{n+1} + C$
$e^{ax}$	$\frac{1}{a}e^{ax} + C$
$a^x$	$\frac{1}{\ln a}a^x + C$
$\frac{1}{x}$	$\ln|x| + C$
$k$ (constant)	$kx + C$

Practice Problems (with Solutions)

• Find the antiderivative: $\int (4x^3 - 2x + 7)\,dx$ Solution: $x^4 - x^2 + 7x + C$

• Find the cost function given marginal cost $C'(x) = 3x^2$ and $C(1) = 10$: $C(x) = x^3 + C$; Use $C(1) = 10$ to solve for $C$: $10 = 1 + C \implies C = 9$; so $C(x) = x^3 + 9$

Key Takeaways

• Antiderivatives reverse the process of differentiation.

• Every function has infinitely many antiderivatives, differing by a constant.

• Indefinite integrals are used to find general solutions; definite integrals (not covered here) are used for specific values.

• Antiderivatives are essential for solving business problems involving accumulation and total change.