Antiderivatives and Indefinite Integration

Antiderivatives

Definition of Antiderivative

An antiderivative of a function f(x) is a function F(x) such that the derivative of F(x) is f(x). In other words,

$F'(x) = f(x)$

The process of finding an antiderivative is called antidifferentiation or indefinite integration.

The Antiderivative and Its Graphical Interpretation

Antiderivatives are not unique; if F(x) is an antiderivative of f(x), then so is F(x) + C, where C is any constant. This is because the derivative of a constant is zero.

• Family of Antiderivatives: All functions of the form F(x) + C are called the general antiderivative of f(x).

• Notation: The general antiderivative is written as $\int f(x)\,dx = F(x) + C$.

Additional info: The image illustrates that the graphs of antiderivatives differ only by a vertical shift (the constant C), but their slopes at any given x are the same.

Indefinite Integrals

Definition and Notation

The indefinite integral of a function f(x) is the set of all its antiderivatives, denoted by:

$\int f(x)\,dx = F(x) + C$

• Integral sign (\int): Indicates the operation of integration.

• Integrand: The function f(x) to be integrated.

• dx: Indicates the variable of integration.

• C: The constant of integration, representing all possible vertical shifts.

Basic Rules of Integration

Power Rule of Integration

For any real number n ≠ -1:

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$

Example: $\int x^2\,dx = \frac{x^{3}}{3} + C$

Constant Multiple Rule

The integral of a constant times a function is the constant times the integral of the function:

$\int k f(x)\,dx = k \int f(x)\,dx$

Sum and Difference Rule

The integral of a sum (or difference) is the sum (or difference) of the integrals:

$\int [f(x) \pm g(x)]\,dx = \int f(x)\,dx \pm \int g(x)\,dx$

Common Indefinite Integrals

Integral of $1/x$

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

Integrals of Exponential Functions

• $\int e^{x}\,dx = e^{x} + C$

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

General Antiderivatives of Trigonometric Functions

• $\int \sin x\,dx = -\cos x + C$

• $\int \cos x\,dx = \sin x + C$

• $\int \sec^2 x\,dx = \tan x + C$

• $\int \csc^2 x\,dx = -\cot x + C$

• $\int \sec x \tan x\,dx = \sec x + C$

• $\int \csc x \cot x\,dx = -\csc x + C$

Finding a Particular Solution: Initial Value Problems

Calculating the Constant of Integration

To find a specific antiderivative (a particular solution), you need an initial value (a point $(x_0, y_0)$ on the function). The steps are:

1. Find the general antiderivative $F(x) + C$.

2. Substitute the initial value into the equation: $y_0 = F(x_0) + C$.

3. Solve for $C$.

4. Write the particular solution using the found value of $C$.

Example: If $\frac{dy}{dx} = 2x$ and $y(1) = 5$, then $y = x^2 + C$. Substitute $x = 1$, $y = 5$: $5 = 1^2 + C \implies C = 4$. So, $y = x^2 + 4$.

Examples

Example 1

Find $\int 3x^2\,dx$.

Solution: $\int 3x^2\,dx = 3 \int x^2\,dx = 3 \left( \frac{x^3}{3} \right) + C = x^3 + C$

Example 2

Find $\int (2x^3 - 5x)\,dx$.

Solution: $\int 2x^3\,dx - \int 5x\,dx = 2 \left( \frac{x^4}{4} \right) - 5 \left( \frac{x^2}{2} \right) + C = \frac{x^4}{2} - \frac{5x^2}{2} + C$

Example 3

Find $\int e^{2x}\,dx$.

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

Example 4

Find $\int \cos x\,dx$.

Solution: $\int \cos x\,dx = \sin x + C$

Example 5

Find the particular solution to $\frac{dy}{dx} = 4x^3$ with $y(1) = 10$.

General solution: $y = x^4 + C$. Substitute $x = 1$, $y = 10$: $10 = 1 + C \implies C = 9$. So, $y = x^4 + 9$.

Example 6

Find $\int \frac{1}{x}\,dx$.

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

Example 7

Find $\int \sin x\,dx$.

Solution: $\int \sin x\,dx = -\cos x + C$

Example 8

Find $\int 7\,dx$.

Solution: $\int 7\,dx = 7x + C$

Example 9

Find $\int (x^2 + 2x + 1)\,dx$.

Solution: $\int x^2\,dx + \int 2x\,dx + \int 1\,dx = \frac{x^3}{3} + x^2 + x + C$

Example 10

Find $\int 5e^{x}\,dx$.

Solution: $5 \int e^{x}\,dx = 5e^{x} + C$