Antiderivatives and Indefinite Integrals

Definition and Basic Properties

The concept of the antiderivative (or integral) is fundamental in calculus. If F(x) is a function whose derivative is f(x), then F(x) is an antiderivative of f(x). The indefinite integral represents the family of all antiderivatives of a function:

• Indefinite Integral:

• Constant of Integration: is an arbitrary constant.

• Integrand: The function being integrated, f(x).

• Differential: indicates the variable of integration.

Integral of a Constant:

Integral of a Power Function: For ,

Integral of a Constant Times a Function:

Integral of a Sum or Difference:

Indefinite Integrals of Exponential Functions

• ,

• ,

• ,

Integration by Substitution

Substitution Method

The substitution method is used to simplify integrals by changing variables. This is especially useful when the integrand is a composite function or involves quotients, exponentials, powers, or radicals.

• Step 1: Make a substitution for x. Let u = some expression in x.

• Step 2: Rewrite the integral in terms of u.

• Step 3: Integrate with respect to u.

• Step 4: Replace u with the original expression in x.

Suggestions for Substitutions:

• If the integrand involves a quotient, let u = denominator.

• If the integrand involves an exponential, let u = exponent.

• If the integrand involves a power or radical, let u = base or radicand.

Definite Integrals and Area

Definition and Properties

The definite integral of a function f(x) over the interval [a, b] is defined as:

The definite integral represents the area under the curve f(x) from x = a to x = b.

Area:

Total Change: If f(x) gives the rate of change of F(x), then the total change in F(x) from a to b is .

Properties of the Definite Integral

• , where k is a constant.

• (No accumulation without change.)

• , where

The Fundamental Theorem of Calculus

Statement and Application

The Fundamental Theorem of Calculus connects differentiation and integration. If F(x) is an antiderivative of f(x) on [a, b], then:

• Lower limit: a; Upper limit: b

Area Between Two Curves

Definition and Calculation

The area bounded by two continuous functions f(x) and g(x) on [a, b], where f(x) is greater than or equal to g(x), is:

Consumer's and Producer's Surplus

Definitions

• Supply Function: p = S(x) gives the price per unit at which producers supply x units.

• Demand Function: p = D(x) gives the price per unit consumers are willing to pay for x units.

Consumer's Surplus

The consumer's surplus is the area between the demand curve and the equilibrium price:

Producer's Surplus

The producer's surplus is the area between the equilibrium price and the supply curve:

Applied Problems

• Marginal Cost: Integrate the marginal cost function to find total cost.

• Revenue Streams: Integrate revenue functions over time to find accumulated revenue.

• Investment and Maintenance: Use definite integrals to calculate total costs or profits over specified intervals.