THE INDEFINITE INTEGRAL AND DIFFERENTIAL EQUATIONS

Flashcards covering the definitions, formulas, and theorems pertaining to indefinite integrals, antiderivatives, and differential equations.

Antiderivative

A function $F$ called an antiderivative (also known as indefinite integral) of the function $f$ on an interval $I$ if $F'(x) = f(x)$ for all $x \text{ in } I$.

Theorem 1.1

If $F$ is a particular antiderivative of $f$ on an interval $I$, then every antiderivative of $F$ is of the form $F(x) + C$, where $C$ is an arbitrary constant.

Antidifferentiation

Also known as integration, it is an operation that finds all antiderivatives of a function and is denoted by the integral sign $\int$.

Constant of Integration

The arbitrary constant $C$ added to the result of an integral, as in $\int f(x)dx = F(x) + C$.

Theorem 1.2 (Power Rule for Integration)

Let $n$ be a rational number such that $n \neq 1$. Then $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for some constant $C$.

$$\int \tan(x)dx$$

$$\ln|\sec(x)| + C$$

$$\int \cot(x)dx$$

$$\ln|\sin(x)| + C$$

$$\int \sec(x)dx$$

$$\ln|\sec(x) + \tan(x)| + C$$

$$\int \csc(x)dx$$

$$\ln|\csc(x) - \cot(x)| + C$$

$$\int a^xdx$$

$\frac{a^x}{\ln(a)} + C$, for $a > 0$ and $a \neq 1$

$$\int e^xdx$$

$$e^x + C$$

$$\int \frac{dx}{x}$$

$\ln|x| + C$, for $x \neq 0$

Integral Constant Multiple Property

$\int c f(x)dx = c \int f(x)dx$ for any constant $c$.

Integral Sum Property

$\int [f(x) + g(x)]dx = \int f(x)dx + \int g(x)dx$.

Theorem 1.4 (Integration of Composite Functions)

Let $g$ be a differentiable function and $F$ be an antiderivative of $f$. Then $\int f(g(x))g'(x)dx = F(g(x)) + C$.

Differential Equation

An equation containing a function and its derivatives, or just derivatives.

Solution to a Differential Equation

A function $f(x)$ satisfying the differential equation for all possible values of the variable $x$.