Antiderivatives and Substitution: Business Calculus Study Notes

Antiderivatives and Indefinite Integrals

Definition and Notation

An antiderivative of a function f(x) is a function F(x) such that F'(x) = f(x) on an interval. The most general antiderivative is called the indefinite integral:

• $\int f(x)\,dx = F(x) + C$, where C is an arbitrary constant.

• For example: $\int 2x\,dx = x^2 + C$

• The dx in the integral indicates integration with respect to x, analogous to $\frac{d}{dx}$ for differentiation.

Basic Rules for Indefinite Integrals

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

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

• Exponential and Logarithmic Functions:

  • $\int e^{kx}\,dx = \frac{e^{kx}}{k} + C$ ($k \neq 0$)

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

  • $\int a^x\,dx = \frac{a^x}{\ln a} + C$ ($a > 0$, $a \neq 1$)

Examples of Basic Integration

• Polynomial: $\int 5x^4\,dx = x^5 + C$

• Term-by-term: $\int (3x^2 - 4x + 5)\,dx = x^3 - 2x^2 + 5x + C$

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

• Exponential: $\int e^{-2x}\,dx = -\frac{1}{2}e^{-2x} + C$

Properties of Antiderivatives

• Any two antiderivatives of a function on an interval differ by a constant.

• If $F(x)$ and $G(x)$ are both antiderivatives of $f(x)$, then $F(x) = G(x) + C$.

Applications of Indefinite Integrals

Revenue and Demand Functions

Indefinite integrals are used to recover total revenue and price functions from marginal revenue.

• Example: Marginal revenue $R'(q) = 4000 - 8q^{-1} + 8q$

• Revenue function: $R(q) = \int R'(q)\,dq = 4000q - 8\ln q + 4q^2 + C$

• Price function: $p(q) = \frac{R(q)}{q}$

• Units: $R(q)$ in dollars, $p(q)$ in dollars per unit, $q$ is quantity.

Motion Problems

Integrals are used to find velocity and position from acceleration.

• Example: An object is thrown downward with $a(t) = -32$ ft/s$^2$, $v(0) = -20$ ft/s, $s(0) = 1100$ ft.

• Velocity: $v(t) = \int a(t)\,dt = -32t + C_1$

• Position: $s(t) = \int v(t)\,dt = -16t^2 - 20t + 1100$

• Time to hit ground: Solve $s(t) = 0$ for $t$ using the quadratic formula.

• Impact speed: $v(t_{impact})$

Substitution Method (Chain Rule in Reverse)

Substitution Formula

Substitution is used to simplify integrals by changing variables.

• If $u = f(x)$ and $du = f'(x)dx$, then:

  • $\int f'(x)[f(x)]^n dx = \int u^n du = \frac{u^{n+1}}{n+1} + C$

  • $\int \frac{f'(x)}{f(x)} dx = \int \frac{du}{u} = \ln|u| + C$

  • $\int f'(x)e^{f(x)} dx = \int e^u du = e^u + C$

Guidelines for Choosing Substitution

• Choose $u$ to be:

  1. The quantity under a root or raised to a power

  2. The quantity in the denominator

  3. The exponent on $e$

• Some integrands may need algebraic rearrangement to fit these cases.

• Match $du$ up to a constant, integrate, and back-substitute to $x$.

Examples of Substitution

• Example 1: $\int 6x(3x^2 + 4)^7 dx$

  • Let $u = 3x^2 + 4$, $du = 6x dx$

  • $\int u^7 du = \frac{u^8}{8} + C$

  • Back-substitute: $\frac{(3x^2 + 4)^8}{8} + C$

• Example 2: $\int x^2 \sqrt{x^3 + 1} dx$

  • Let $u = x^3 + 1$, $du = 3x^2 dx$

  • Multiply and divide by 3: $\frac{1}{3} \int \sqrt{u} du$

  • Integrate: $\frac{1}{3} \cdot \frac{2}{3} u^{3/2} + C$

  • Back-substitute: $\frac{2}{9}(x^3 + 1)^{3/2} + C$

• Example 3: $\int \frac{x+3}{x^2 + 6x} dx$

  • Let $u = x^2 + 6x$, $du = 2x + 6 dx$

  • Express $dx$ in terms of $du$ and $x$

  • Integrate: $\frac{1}{2} \int u^{-1} du = \frac{1}{2} \ln|u| + C$

  • Back-substitute: $\frac{1}{2} \ln|x^2 + 6x| + C$

• Example 4: $\int \frac{2x-3}{x^2-3x} dx$

  • Let $u = x^2 - 3x$, $du = 2x - 3 dx$

  • Integrate: $\int \frac{du}{u} = \ln|u| + C$

  • Back-substitute: $\ln|x^2 - 3x| + C$

• Example 5: $\int 3x^2 e^{x^3} dx$

  • Let $u = x^3$, $du = 3x^2 dx$

  • Integrate: $\int e^u du = e^u + C$

  • Back-substitute: $e^{x^3} + C$

• Example 6: $\int x \sqrt{1-x} dx$

  • Let $u = 1 - x$, $du = -dx$, $x = 1 - u$

  • Substitute and integrate: $-\int (1-u)u^{1/2} du$

  • Expand and integrate: $-\int u^{1/2} du + \int u^{3/2} du$

  • Back-substitute: $\frac{2}{3}(1-x)^{3/2} - \frac{2}{5}(1-x)^{5/2} + C$

Summary Table: Common Indefinite Integrals

Function	Indefinite Integral
$x^n$ ($n \neq -1$)	$\int x^n dx = \frac{x^{n+1}}{n+1} + C$
$e^{kx}$	$\int e^{kx} dx = \frac{e^{kx}}{k} + C$
$\frac{1}{x}$	$\int \frac{1}{x} dx = \ln|x| + C$
$a^x$	$\int a^x dx = \frac{a^x}{\ln a} + C$

Key Points for Business Calculus Students

• Antiderivatives are essential for solving problems involving accumulation, such as total revenue, position, and demand.

• Substitution is a powerful technique for integrating composite functions, especially those involving powers, roots, denominators, or exponents.

• Always check your work by differentiating your answer to recover the original integrand.