Antiderivatives and Indefinite Integrals
Define an antiderivative F of f where F'= f. Build the complete table of elementary indefinite integrals. Understand the +C constant of integration and apply initial conditions to find particular antiderivatives.
A lesson of AP Calculus AB, in Integration and the Fundamental Theorem of Calculus, about 10 minutes.
Antiderivatives and Definite Integrals
By the end of this lesson, you will be able to find antiderivatives using basic rules and substitution, evaluate definite integrals, and apply the Fundamental Theorem of Calculus.
Antiderivatives (Indefinite Integrals)
Antiderivative: a function $F$ such that $F'(x) = f(x)$.
The indefinite integral represents the family of all antiderivatives:
$$\int f(x), dx = F(x) + C$$
where $C$ is the constant of integration (since derivatives of constants are 0, any $F(x) + C$ is a valid antiderivative).
Basic Antiderivative Rules
Power Rule for Integration (for $n \neq -1$):
$$\int x^n, dx = \frac{x^{n+1}}{n+1} + C$$
Note: this is the "undo" of the power rule for differentiation.
Examples:
$\int x^4, dx = \dfrac{x^5}{5} + C$
$\int \sqrt{x}, dx = \int x^{1/2}, dx = \dfrac{x^{3/2}}{3/2} + C = \dfrac{2}{3}x^{3/2} + C$
$\int \dfrac{1}{x^2}, dx = \int x^{-2}, dx = \dfrac{x^{-1}}{-1} + C = -\dfrac{1}{x} + C$
Key antiderivatives to memorize:
| $f(x)$ | $\int f(x), dx$ |
|---|---|
| $\dfrac{1}{x}$ | $\ln |
| $e^x$ | $e^x + C$ |
| $a^x$ | $\dfrac{a^x}{\ln a} + C$ |
| $\sin x$ | $-\cos x + C$ |
| $\cos x$ | $\sin x + C$ |
| $\sec^2 x$ | $\tan x + C$ |
| $\csc^2 x$ | $-\cot x + C$ |
| $\sec x \tan x$ | $\sec x + C$ |
The Fundamental Theorem of Calculus
This is the central theorem connecting differentiation and integration.
Part 1 (FTC Part 1):
If $F(x) = \displaystyle\int_a^x f(t), dt$ (where $f$ is continuous), then:
$$F'(x) = \frac{d}{dx}\int_a^x f(t), dt = f(x)$$
The derivative of an integral (with variable upper limit) is the integrand.