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Antiderivatives & Indefinite Integrals

Antiderivatives

  1. An antiderivative of a function is a function whose derivative is the original function. The process of finding an antiderivative is called antidifferentiation.

  • Definition: If , then is an antiderivative of .

Notation: The general antiderivative of is , where is an arbitrary constant.

Key Table:

Function

Derivative

Antiderivative

$2$

  • Example: Find the antiderivative of .

    • Solution:

  • Example: Find the antiderivative of .

    • Solution:

Additional info: Always check your answer by differentiating your result to see if you recover the original function.

Finding a Particular Antiderivative

Given a specific point, we can determine the constant to find a unique antiderivative that passes through that point.

  1. General Solution:

  2. Particular Solution: Use the given condition to solve for .

  3. Example: Find the antiderivative of given .

    • General antiderivative:

    • Plug in :

    • Particular solution:

Indefinite Integrals

Introduction to Indefinite Integrals

The indefinite integral of a function is the set of all its antiderivatives, denoted by .

  • Notation:

  • Meaning: The process of integration is the reverse of differentiation.

  • Example:

Power Rule for Indefinite Integrals

The power rule allows us to integrate functions of the form (where ):

Form

Rule

Example

  • Example:

Additional Rules for Indefinite Integrals

Integration is linear, so the following rules apply:

Name

Rule

Example

Sum & Difference

Constant Multiple

Integrals of Trigonometric Functions

Integrals Resulting in Basic Trig Functions

Integrating sine and cosine functions yields:

Integral

Result

  • Example:

Initial Value Problems

Solving Initial Value Problems (IVPs)

An initial value problem involves finding a particular solution to a differential equation that satisfies a given initial condition.

  • General Steps:

    1. Find the general antiderivative (integrate the function).

    2. Apply the initial condition to solve for .

    3. Write the particular solution.

  • Example: Solve , .

    • Integrate:

    • Apply :

    • Solution:

Additional info: For higher-order differential equations, integrate repeatedly and apply all given initial conditions.

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