NCERT Class 12 Mathematics Mathematics Part-II: Chapter 1 — INTEGRALS
This chapter introduces Integral Calculus, which is motivated by finding the area under curves and determining a function given its derivative. It explains integration as the inverse process of differentiation, where finding the original function from its derivative is called finding the anti-derivative or integral. The chapter highlights that anti-derivatives are not unique and involve an arbitrary constant of integration (C). It establishes the connection between indefinite and definite integrals through the Fundamental Theorem of Calculus, noting their importance in science, engineering, economics, finance, and probability. The focus is on understanding indefinite and definite integrals, their properties, and basic integration techniques, crucial for advanced mathematics and problem-solving in CBSE Class 12.
Quick info
| Board | CBSE / NCERT |
|---|---|
| Class | Class 12 |
| Subject | Mathematics |
| Book | Mathematics Part-II |
| Chapter | Chapter 1 — INTEGRALS |
| Language | English |
| PDF type | NCERT Textbook |
| Session | CBSE 2026 |
| Reading time | 4 minutes |
| Word count | 711 |
Learning outcomes
- Understand integration as the inverse process of differentiation.
- Identify and find anti-derivatives (integrals) of functions.
- Recognize the role of the arbitrary constant of integration.
- Understand the basic concepts of indefinite and definite integrals.
- Appreciate the connection between differentiation and integration (Fundamental Theorem of Calculus).
Vocabulary
| Word | Meaning |
|---|---|
| Derivative | The rate of change of a function with respect to a variable. |
| Integral Calculus | The branch of calculus concerned with the properties and applications of integrals. |
| Anti derivative | A function whose derivative is a given function. |
| Indefinite integral | The set of all anti derivatives of a function, including an arbitrary constant. |
| Definite integral | An integral that evaluates to a specific numerical value, representing an area or accumulation. |
| Constant of integration | The arbitrary constant 'C' added to the indefinite integral. |
| Fundamental Theorem of Calculus | Connects differentiation and integration, showing definite integrals can be evaluated using anti-derivatives. |
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Practice questions
- What is integration considered as in relation to differentiation? Answer: Integration is considered the inverse process of differentiation.
- What is another name for an anti-derivative? Answer: An integral.
- Why is an arbitrary constant 'C' added to an indefinite integral? Answer: Because the derivative of a constant is zero, so there are infinitely many anti-derivatives for a given function.
- What are the two main problems that led to the development of integral calculus? Answer: Finding a function given its derivative, and finding the area bounded by the graph of a function.
Practice MCQs
Q1. Integral Calculus is motivated by the problem of defining and calculating:
Explanation: Integral calculus is primarily concerned with finding areas under curves and volumes, which is directly related to calculating the area of a region bounded by functions.
Q2. The process of finding an anti-derivative is called:
Explanation: Integration is defined as the inverse process of differentiation, where we find the original function from its derivative, also known as finding the anti-derivative.
Q3. If F(x) is an anti-derivative of f(x), then the indefinite integral of f(x) is:
Explanation: The indefinite integral of a function f(x) is F(x) + C, where F(x) is any particular anti-derivative and C is the arbitrary constant of integration.
Q4. The constant added to an indefinite integral is called:
Explanation: This constant, denoted by 'C', is referred to as the arbitrary constant or the constant of integration because it can take any real value.
Q5. Which theorem connects indefinite and definite integrals?
Explanation: The Fundamental Theorem of Calculus establishes a crucial link between differentiation and integration, enabling the evaluation of definite integrals using anti-derivatives.
Frequently asked questions
What is the primary motivation behind Integral Calculus?
Integral Calculus is motivated by the problem of defining and calculating the area of regions bounded by the graphs of functions, and by the problem of finding a function given its derivative.
What does it mean for integration to be the inverse process of differentiation?
It means that if you differentiate a function to get its derivative, integration allows you to find the original function from its derivative.
Why do indefinite integrals always include a '+ C'?
The '+ C' represents the constant of integration. Since the derivative of any constant is zero, there are infinitely many functions that have the same derivative, differing only by a constant.
What is the role of the Fundamental Theorem of Calculus?
It connects indefinite and definite integrals, providing a method to calculate definite integrals using anti-derivatives, making it a practical tool in science and engineering.
What is the difference between an indefinite and a definite integral?
An indefinite integral represents a family of functions (anti-derivatives) and includes an arbitrary constant 'C', while a definite integral evaluates to a specific numerical value, often representing an area.
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NCERT Class 12 Mathematics — Mathematics Part-II — Chapter 1 — INTEGRALS. Verified by NCERT Help Editorial Team. Reviewed on 29 Jul 2026. Last updated 10 Aug 2026.