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📘 Antiderivative
Grade 12 · Mathematics · Chapter: Antiderivatives (Integration)
Learning Objectives
- Understand the meaning and definition of an antiderivative (indefinite integral).
- Recognize integration as the inverse process of differentiation.
- Apply the family-of-antiderivatives concept, \(\int f(x)\,dx = F(x)+C\).
- Evaluate indefinite integrals using algebraic simplification, trigonometric identities, and rationalization.
- Connect antiderivatives to real-life applications (revenue, distance, structural stress).
- Solve worked examples, exercises, quizzes and a full CDC-model examination.
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🚀 Real-Life Applications of Integration
Integration is the mathematical process of finding the total, or accumulating small parts. Here are three real-world examples where integration (antiderivatives) is used every day.
Economics — Total Revenue
Problem: A company's sales rate fluctuates hourly based on promotions, web traffic, and time zones.
Integration fix: Integrating the changing rate of sales over a 24-hour period gives the exact total revenue earned that day.
Technology — Fitness Trackers
Problem: A phone's accelerometer only measures changing speed and acceleration moment by moment — not total distance.
Integration fix: The fitness app integrates speed over time to calculate the total distance covered.
Engineering — Safe Bridges
Problem: The weight of cars, wind pressure, and steel beams creates a variable force across a bridge's length.
Integration fix: Engineers integrate the force distribution function across the bridge's shape to calculate total stress.
📊 Visualization: Area, Accumulation & the Antiderivative
Drag the slider to change the number of trapezoids used to approximate the area under \(f(x)=2x\) from 0 to x. Watch the approximate area converge to the exact antiderivative value \(F(x)\) — this is the link between accumulation and the antiderivative.
📖 Mathematical Definition of an Antiderivative
An antiderivative (or primitive function) of a function \(f(x)\) is a function \(F(x)\) whose derivative is equal to \(f(x)\) on a given interval.
then \(F(x)\) is called an antiderivative of \(f(x)\).
Since the derivative of a constant is zero, every antiderivative differs by a constant. Therefore, the family of all antiderivatives is written as:
where:
- \(\int\) is the integral sign,
- \(f(x)\) is the given function (integrand),
- \(dx\) indicates integration with respect to \(x\),
- \(F(x)\) is an antiderivative of \(f(x)\),
- \(C\) is an arbitrary constant of integration.
Example
Since \(\dfrac{d(x^2)}{dx} = 2x\), therefore \(\int 2x\,dx = x^2 + C\).
Hence, \(x^2+C\) is the family of antiderivatives of \(2x\).
Quick Check
🧮 Formula Derivation — Rules of Integration
These general rules are the tools used throughout the Examples, Exercises and Quizzes in this chapter.
✏️ Worked Out Examples
🧠 Quizzes (25 Questions)
Quizzes are locked. Tap 🔒 Unlock above and enter the password to view answers & explanations for the whole session.
📝 Exercise – 1
Q2. Evaluate the following integrals
Q3. Evaluate the following integrals
📝 Exercise – 2
Q4. Evaluate the following integrals
Q5. Evaluate the following integrals
🎓 Examination (CDC Nepal Model)
5 questions × 2 marks + 5 questions × 3 marks + 5 questions × 5 marks = Full Marks 50
✍️ Answer Sheet
Type LaTeX (e.g. \int x^2\,dx) or use the symbol buttons:
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