An overview of The Integrals Chapter
Integrals are mathematical objects, which can be interpreted as a generalisation of an area. Moreover, coupled with derivatives, they form the fundamental of calculus. Integrals are divided into two subtypes, definite integrals and indefinite integrals.
Additionally, every definition of integrals has a perfect measure. It is the base of integration. For instance, Riemann integral is based on Jordan Measure. Similarly, the Lebesgue integral is based on Lebesgue measure. Moreover, the Riemann integral is regarded as the simplest integral definition and used in physics and calculus.
Math class 12 chapter 7 integrals
In this chapter, you will study about integrals and their applications.
7.1- Introduction
At the beginning of the chapter, you are introduced to the concept of integration meaning, and its history. Moreover, you also learn about definite and indefinite integrals. Furthermore, you are introduced to the fundamental concept of indefinite integrals. Integration is also regarded as anti-derivative.
7.2- Integration As An Inverse Process Definition
In this section of integrals, you will learn how to use the inverse process of differentiation. Instead of differentiating a function, you will get a derivative of a particular function, after that, you need to find the primitive function. This process of calculating integrals is regarded as the inverse process. There are several integration examples to help you comprehend the topic even better.
7.3- Methods of Integration
You have already learnt about the integrals functions of which are readily obtained from derivatives. Additionally, it was based on inspection, i.e. searching functions which derivatives will lead to integrals. However, this section deals with a higher level of sums that require different methods like –
Integration via submission.
Integration by partial functions.
integration by parts.
Additionally, there is an exercise with different questions which require critical thinking for solving these problems.
7.4- Integrals of Some Particular Functions
As you have already studied about trigonometric functions like sine, tangent, etc. In this section, you use those functions to build formulas, which will help you during an exam.
7.5- Integration by Partial Fractions
The focus point of this section, disintegrating equations into parts and discovering the integral of every section. This partial disintegration process makes the overall approach easier. Additionally, it proves highly efficient in finding a solution to any problem.
7.6- Integration by Parts
This section is important for understanding the processes of solving problems. Moreover, in this portion, you learn more than one method of integration. Hence, these methods are useful in integrating products of functions. However, you should know that it does not apply to a product of functions in every case.
7.7- Definite Integral
In the previous sections, you have already covered indefinite integrals and methods to find them. Moreover, it also included discussion of some special integrals. In this section, you will learn about definite integrals.
Furthermore, definite integrals have a unique value. It is either introduced as the ‘limit of a sum’ or if it has an anti-derivative ‘F’ at its endpoints.
7.8- Fundamental Theorem of Calculus
This section of integrals is very interesting as it combines both mensuration and calculus. Therefore, it helps in finding areas covered under curves denoted by coordinates. Moreover, you get a chance to study fundamental theorems in this chapter.
7.9- Evolution of Definite Integrals by Substitution
Integration by substitution is a vital method to determine a definite integral of a function. You have already learnt about other methods in the previous sections. Moreover, substitution method also has a similar approach like others. However, you should practice every method.
7.10- Some Properties of Definite Integrals
In this section, you will learn about various essential properties of definite integrals. Furthermore, knowing these properties will help you evaluate definite integrals.
7.11- Exercise
In this exercise section, you get a chance to revise what you have studied till now. The questions are all based on concepts of integration.
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Integrals is an important chapter of class 12 mathematics. Moreover, it helps you to learn the fundamentals of this topic and then moves on to the advance sections. Additionally, you can visit the website as well as the app of Vedantu to access various study guides for this chapter.
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FAQs on Integrals
1. What is the meaning of integration in Class 12 Maths, and how is it connected to finding areas under curves?
Integration in Class 12 Maths refers to the process of determining the antiderivative of a function. It is used to calculate the area under a curve between defined limits, which is essential for solving problems related to distance, area, and other applications in calculus as per the CBSE 2025–26 syllabus.
2. How do definite and indefinite integrals differ as per the CBSE Class 12 Integrals chapter?
A definite integral provides a specific numerical value representing the area between two limits, while an indefinite integral gives a general formula for the antiderivative and includes a constant of integration (C). The choice depends on whether fixed boundaries are given in the question.
3. What are the official methods of integration students must master for the 2025–26 CBSE board exam?
The key methods of integration according to the Class 12 syllabus include:
- Integration by substitution
- Integration by partial fractions
- Integration by parts
- Direct use of standard integrals for specific function types
4. Why is integration called the inverse process of differentiation?
Integration reconstructs the original function from its derivative, reversing the operation of differentiation. This is why integration is often termed anti-differentiation in calculus.
5. How do you choose the correct method of integration for a specific Class 12 question?
Select the integration technique based on the structure of the integrand:
- If the expression includes a function and its derivative, use substitution.
- For rational functions with factorable denominators, prefer partial fractions.
- For products of different types of functions (e.g. algebraic with trigonometric), apply integration by parts.
6. What common mistakes should be avoided in solving definite integrals in CBSE board exams?
Common traps include:
- Forgetting to apply upper and lower limits correctly after integrating
- Assuming the definite integral always gives a positive area (it can be negative if the curve is below the x-axis)
- Omitting the evaluation steps required for full marks under CBSE marking schemes
7. Can you explain how the Fundamental Theorem of Calculus bridges differentiation and integration?
The Fundamental Theorem of Calculus states that differentiation and integration are opposite operations. It allows you to evaluate a definite integral by finding an antiderivative and subtracting its values at the given limits, making calculation efficient and conceptually clear in all board exam questions.
8. How do integrals help solve real-world problems in subjects like Physics and Economics?
Integrals are important for determining:
- Areas under curves (physics, geography)
- Volumes of solids (engineering, construction)
- Work done by variable forces (physics)
- Probability calculations in statistics and economics
9. What effective strategies can students use in the board exam to maximize marks in integrals questions?
Boost exam marks by:
- Identifying the right integration method for each question
- Showing clear step-by-step calculations
- Stating the constant of integration where required
- Highlighting the final answer
- Not skipping any steps, as CBSE awards method marks
10. Why is understanding integrals considered essential for advanced studies and career choices in science and mathematics?
Integrals form the basis for more complex processes like solving differential equations, studying real-world physical systems, and analyzing variable rates. Mastering them is crucial for success in engineering, physics, economics, and data science.
11. What roles do standard integrals and integration formulas play in the Class 12 board exam?
Standard integrals and formulas help solve integration problems more efficiently by providing pre-derived results for commonly used functions, reducing calculation time and minimizing errors during the exam as per the official CBSE methods.
12. How does partial fraction decomposition simplify integration problems involving rational functions?
Partial fraction decomposition breaks rational expressions into simpler terms, making them easier to integrate individually. This method is especially useful when denominators factor into distinct linear or quadratic expressions, as prescribed in the CBSE syllabus.
13. Explain a scenario where integration by substitution is the most efficient method and why.
Integration by substitution is most efficient when the integrand includes a function and its derivative. For example, integrating (2x)·cos(x²) dx is simplified by setting u = x², making the process direct and minimizing errors.
14. What misconceptions might students have about the constant of integration in indefinite integrals?
Many students think the constant of integration, C, is optional. However, omitting it leads to incomplete answers and possible loss of marks, since indefinite integrals can represent a family of functions.
15. In which types of Class 12 board questions is integration by parts the required method, and why?
Integration by parts is necessary when the integrand is a product of two distinct function types, such as an algebraic and a trigonometric function. CBSE papers often test this method, so recognizing such scenarios ensures students apply the correct steps for full marks.
