Antiderivatives and Indefinite Integrals
Define antiderivatives and the indefinite integral ∫f(x)dx = F(x) + C. Apply the power rule for integration, and build the table of basic integral formulas (trig, exponential, logarithmic).
A lesson of Calculus, in Integration and the Fundamental Theorem, about 10 minutes.
By the end of this lesson, you will be able to find antiderivatives of elementary functions, apply the Power Rule for integration, and express the general antiderivative with the constant of integration $C$.
The Big Idea
An antiderivative of $f$ is a function $F$ such that $F'(x) = f(x)$. Finding antiderivatives is the "reverse" of differentiation. This process, called indefinite integration, gives a family of functions (differing by a constant $C$) rather than a unique function.
The Integral Notation
$$\int f(x),dx = F(x) + C$$
Read: "the integral of $f(x)$ with respect to $x$ equals $F(x)$ plus $C$."
- $\int$: the integral sign
- $f(x)$: the integrand
- $dx$: indicates the variable of integration
- $F(x) + C$: the general antiderivative ($C$ is an arbitrary constant)
Basic Antiderivative Rules
Power Rule for Integration (Power Rule Reversed):
$$\int x^n,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$
Constant Rule: $\int k,dx = kx + C$
Sum/Difference: $\int [f(x) \pm g(x)],dx = \int f(x),dx \pm \int g(x),dx$
Constant Multiple: $\int kf(x),dx = k\int f(x),dx$
Standard Antiderivatives
| $f(x)$ | $\int f(x),dx$ |
|---|---|
| $x^n$ ($n \neq -1$) | $\frac{x^{n+1}}{n+1} + C$ |
| $\frac{1}{x}$ | $\ln |
| $e^x$ | $e^x + C$ |
| $a^x$ | $\frac{a^x}{\ln a} + C$ |
| $\sin x$ | $-\cos x + C$ |
| $\cos x$ | $\sin x + C$ |
| $\sec^2 x$ | $\tan x + C$ |
| $\csc^2 x$ | $-\cot x + C$ |
| $\sec x\tan x$ | $\sec x + C$ |
| $\csc x\cot x$ | $-\csc x + C$ |
| $\frac{1}{\sqrt{1-x^2}}$ | $\sin^{-1}x + C$ |
| $\frac{1}{1+x^2}$ | $\tan^{-1}x + C$ |