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Integration & Antiderivatives

Emily Davis

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

  1. Introduction
  2. Finding the Constant of Integration
  3. Worked Example
  4. Practice Questions
  5. Glossary
  6. Summary and Key Takeaways
  7. 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

Key Concept

When finding an indefinite integral, a constant of integration is needed. The general form of an indefinite integral is:

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

where F(x)=f(x)F'(x) = f(x) and CC is any constant.

Steps to Find the Constant

  1. Integrate the Function:
    • Perform the integration to find the general form of the antiderivative.
  2. Use Additional Information:
    • If additional information is provided, such as a specific value of F(x)F(x) for a given xx, or a point (x0,y0)(x_0, y_0) that the graph passes through, use this to find CC.
  3. Set Up and Solve an Equation:
    • Use the given point to set up an equation and solve for CC.

Example Process

Suppose we need to integrate f(x)=2x+3f(x) = 2x + 3 and find the constant of integration given that F(x)F(x) goes through the point (1,2)(1, 2).

  1. Integrate: F(x)=(2x+3),dx=x2+3x+CF(x) = \int (2x + 3) , dx = x^2 + 3x + C

  2. Use Given Point:

    • Given F(1)=2F(1) = 2, substitute x=1x = 1 and F(x)=2F(x) = 2 into the equation: 2=12+3(1)+C2 = 1^2 + 3(1) + C
  3. Solve for CC: 2=1+3+C2=4+CC=22 = 1 + 3 + C \Rightarrow 2 = 4 + C \Rightarrow C = -2

Thus, the specific antiderivative is: F(x)=x2+3x2F(x) = x^2 + 3x - 2

Exam Tip

Always check if additional conditions are provided to find the exact value of the constant of integration.

Common Mistake

Students often forget to include the constant of integration when finding indefinite integrals. Always remember to add CC after integrating.

Worked Example

The function hh measures the height of a projectile above the ground at time tt (in seconds). It is known that hh satisfies the equation:

h(t)=(7032t),dth(t) = \int (70 - 32t) , dt

and at time t=2t = 2, the projectile is 81 feet above the ground.

Find an explicit expression for hh in terms of tt.

Solution

  1. Integrate: h(t)=(7032t),dt=70t16t2+Ch(t) = \int (70 - 32t) , dt = 70t - 16t^2 + C

  2. Use Given Point:

    • Given h(2)=81h(2) = 81, substitute t=2t = 2 and h(t)=81h(t) = 81 into the equation: 81=70(2)16(2)2+C81 = 70(2) - 16(2)^2 + C
  3. Solve for CC: 81=14064+C81=76+CC=581 = 140 - 64 + C \Rightarrow 81 = 76 + C \Rightarrow C = 5

Thus, the explicit expression for hh is: h(t)=70t16t2+5h(t) = 70t - 16t^2 + 5

Practice Questions

Practice Question
  1. Find the constant of integration for the integral (4x7),dx\int (4x - 7) , dx given that the antiderivative passes through the point (2,1)(2, 1).
Practice Question
  1. Integrate the function f(x)=3x22x+1f(x) = 3x^2 - 2x + 1 and determine the constant of integration if the function passes through the point (1,4)(1, 4).

Glossary

  • Constant of Integration (CC): 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, CC.
  • Use additional information, such as specific points or values, to solve for CC.
  • The process of finding CC 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 CC.
  • Use Given Points Accurately: Substitute them correctly into the integrated function to find CC.

By following these strategies and understanding the process, you'll be better prepared to handle questions involving constants of integration in your exams.

Question 1 of 5

Why do we add a constant of integration, CC, 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