Lesson 3: Introduction to Integration

👨🏽‍🏫 Introduction

Have you ever tried to find how much water fills a bucket, how far a car travels when its speed is changing, or the area under a curve? That’s where integration comes in! 📐🪣🚗

While differentiation helps us find rates of change, integration does the reverse — it helps us find total quantities like area, distance, and volume.

• 🧠 Understand what integration means and how it relates to differentiation
• 🔢 Learn the basic rules of indefinite integrals
• 📐 Find general solutions to simple differential expressions

🧠 Key Concepts

📌 1. What Is Integration?

• Integration is the process of finding the antiderivative or the original function from its derivative.
• It is the reverse of differentiation.
• The symbol for integration is:
  

📌 2. Indefinite Integrals

An indefinite integral does not include limits. It represents a general function plus a constant of integration (+ C) because derivatives of constants are zero.

Basic Rule:

where

Examples:

📌 3. Common Integrals to Remember

🔍 Worked Examples

1. 1. Example 1 – Basic Power Rule:
      Find
      .
      

1. 1. Example 2 – Sum of Terms:
      Find
      .
      

1. 1. Example 3 – Exponential Function:
      Find
      .
      

1. Example 4 – Logarithmic Integral:
   Find
   .
   

✍🏽 Practice Exercises

🅰️ Section A – Basic Indefinite Integrals

🅱️ Section B – Common Functions

🅾️ Section C – Application Thought

1. The velocity of a car is
   .
   Find the displacement after 5 seconds by integrating.

✅ Answers and Explanations

• A1:
  
• A2:
  
• B1:
  
• B2:
  
• C1:
  Displacement =
  
  At
  :
  .
  🟢 Answer: Displacement = 150 units

📌 Recap

• Integration is the reverse of differentiation – it rebuilds functions from their derivatives.
• The symbol
  
  is used for integration.
• Indefinite integrals give a general solution with + C.
• We use integration to find area, displacement, total value, and more.

💭 Reflection Prompt

Have you ever tracked your daily steps or water usage over time? If you knew how fast the amount was changing at each point, could you calculate the total? That’s integration! 📈💧👟