Integration & Antiderivatives

Emily Davis
5 min read
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Study Guide Overview
This study guide covers the constant of integration, focusing on its importance in indefinite integrals. It outlines the steps to find the constant using provided points and demonstrates the process with a worked example. It also includes practice questions, a glossary, key takeaways, and exam strategies related to this concept.
Table of Contents
- Introduction
- Finding the Constant of Integration
- Worked Example
- Practice Questions
- Glossary
- Summary and Key Takeaways
- Exam Strategy
Introduction
In calculus, finding the constant of integration is a crucial step when solving indefinite integrals. This process allows us to determine the specific form of an antiderivative that fits given conditions.
Finding the Constant of Integration
Key Concepts
When finding an indefinite integral, a constant of integration is needed. The general form of an indefinite integral is:
where and is any constant.
Steps to Find the Constant
- Integrate the Function:
- Perform the integration to find the general form of the antiderivative.
- Use Additional Information:
- If additional information is provided, such as a specific value of for a given , or a point that the graph passes through, use this to find .
- Set Up and Solve an Equation:
- Use the given point to set up an equation and solve for .
Example Process
Suppose we need to integrate and find the constant of integration given that goes through the point .
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Integrate:
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Use Given Point:
- Given , substitute and into the equation:
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Solve for :
Thus, the specific antiderivative is:
Always check if additional conditions are provided to find the exact value of the constant of integration.
Students often forget to include the constant of integration when finding indefinite integrals. Always remember to add after integrating.
Worked Example
The function measures the height of a projectile above the ground at time (in seconds). It is known that satisfies the equation:
and at time , the projectile is 81 feet above the ground.
Find an explicit expression for in terms of .
Solution
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Integrate:
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Use Given Point:
- Given , substitute and into the equation:
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Solve for :
Thus, the explicit expression for is:
Practice Questions
Practice Question
- Find the constant of integration for the integral given that the antiderivative passes through the point .
Practice Question
- Integrate the function and determine the constant of integration if the function passes through the point .
Glossary
- Constant of Integration (): An arbitrary constant added to the antiderivative of a function when computing an indefinite integral.
- Indefinite Integral: An integral without upper and lower limits, representing a family of functions.
- Antiderivative: A function whose derivative is the given function.
Summary and Key Takeaways
- When finding an indefinite integral, always include the constant of integration, .
- Use additional information, such as specific points or values, to solve for .
- The process of finding is equivalent to finding the particular solution to a differential equation.
Exam Strategy
- Read the Problem Carefully: Ensure you understand all given conditions and points.
- Double-Check Integration: Make sure your integration is correct and includes .
- Use Given Points Accurately: Substitute them correctly into the integrated function to find .
By following these strategies and understanding the process, you'll be better prepared to handle questions involving constants of integration in your exams.

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Question 1 of 5
Why do we add a constant of integration, , when finding an indefinite integral? 🤔
To make the integral definite
Because the derivative of a constant is always zero
To simplify the integration process
Because all functions have a constant term