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NCERT Solutions for Class 12 Maths Chapter 7 Integrals

NCERT Solutions for Class 12 Maths Chapter 7 has been given in an easy-to-understand way for CBSE Class 12 Students. Students can find the solutions to Maths Chapter 7 Integrals on this page!
authorImageKrati Saraswat21 Jan, 2024
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NCERT Solutions for Class 12 Maths Chapter 7

NCERT Solutions for Class 12 Maths Chapter 7 Integrals have been created by subject experts to aid students in their board exam preparations. The Class 12 NCERT Maths Book encompasses the concept of integrals in Chapter 7. In this section of NCERT Solutions for Class 12 Maths chapter 7, students learn about integral calculus (definite and indefinite), its properties, and more. The topic is crucial for both the CBSE board exam and competitive exams.

The integrals presented in this chapter are detailed and explained in an easy-to-understand manner. These NCERT Solutions for Class 12 Maths chapter 7 integrals are simple and assist students in comprehending problem-solving methods effortlessly. Students can access these NCERT Solutions for Class 12 Maths Chapter 7 for free and download them for offline practice as well. These solved questions guide students in understanding how to apply integral concepts to solve problems effectively.

CBSE Class 12 Maths Syllabus

NCERT Solutions for Class 12 Maths Chapter 7 Overview

A quick overview of Class 12 Maths Chapter 7 in the NCERT textbook reveals a focus on integrals. In this chapter, students delve into integral calculus, covering both definite and indefinite integrals. The emphasis is on understanding the properties of integrals and their significance. This knowledge is crucial for success in CBSE board exams and competitive tests. The chapter provides detailed and easily understandable explanations of integral concepts. These NCERT Solutions for Class 12 Maths Chapter 7 offer a simple approach to problem-solving, allowing students to grasp the application of integral concepts effectively. Accessible for free, students can utilise these solutions online or download them for offline practice, aiding in a comprehensive understanding of the chapter.

NCERT Solutions for Class 12 Maths Chapter 1

NCERT Solutions for Class 12 Maths Chapter 7 Chapter at a Glance

7.1 Introduction: Important Points:
  • Introduction to integration as the inverse of differentiation.
  • Understanding the notion of accumulation of quantities over an interval.
  • Overview of definite and indefinite integrals.
7.2 Integration as an Inverse Process of Differentiation: Important Points:
  • Recognizing integration as the reverse process of differentiation.
  • Understanding the connection between rates of change and accumulated change.
Formulas: If F'(x) = f(x), then ∫ f(x) dx = F(x) + C, where C is the constant of integration.

NCERT Solutions for Class 12 Maths Chapter 2

7.3 Geometrical Interpretation of Definite Integral: Important Points:
  • Geometric interpretation of definite integrals as the signed area under a curve.
  • Positive and negative areas representing different quantities.
Formulas: ∫_a^b f(x) dx represents the signed area between the curve y = f(x) and the x-axis from x = a to x = b. 7.4 Definition of Indefinite Integral and Some of its Properties: Important Points:
  • Introduction to the concept of the indefinite integral.
  • Understanding the indefinite integral as a family of functions.
Formulas:

NCERT Solutions for Class 12 Maths Chapter 3

∫ f(x) dx = F(x) + C, where F(x) is the antiderivative of f(x) and C is the constant of integration. 7.5 Comparison between Differentiation and Integration: Important Points:
  • Highlighting the inverse relationship between differentiation and integration.
  • Introduction to the fundamental theorem of calculus.
Formulas: Fundamental Theorem of Calculus: If F(x) is an antiderivative of f(x), then ∫_a^b f(x) dx = F(b) - F(a). 7.6 Standard Integrals: Important Points:
  • Introduction to a set of standard integrals for elementary functions.
  • Memorising common integrals for efficient problem-solving.
Formulas: Various standard integrals, such as ∫ x^n dx, ∫ e^x dx, ∫ sin x dx, ∫ cos x dx, etc.

NCERT Solutions for Class 12 Maths Chapter 4

7.7 Evaluation of Definite Integrals by Substitution: Important Points:
  • Introduction to the technique of substitution to simplify integrals.
  • Using a change of variables to make integrals more manageable.
Formulas: ∫_a^b f(g(x)) ⋅ g'(x) dx = ∫_{g(a)}^{g(b)} f(u) du (Substitution formula). 7.8 Fundamental Theorem of Calculus: Important Points:
  • Understanding the fundamental theorem's role in connecting definite integrals and antiderivatives.
  • Realising the importance of antiderivatives in evaluating definite integrals.
Formulas: Fundamental Theorem of Calculus: If F(x) is an antiderivative of f(x), then ∫_a^b f(x) dx = F(b) - F(a).

NCERT Solutions for Class 12 Maths Chapter 5

7.9 Evaluation of Definite Integrals by Parts: Important Points:
  • Introduction to the technique of integration by parts.
  • Applying integration by parts to integrals involving the product of two functions.
Formulas: ∫ u dv = uv - ∫ v du (Integration by Parts formula). 7.10 Solved Examples: Important Points:
  • Application of various integration techniques in solving real-world problems.
  • Understanding the step-by-step process of solving integration problems.
7.11 Exercise Questions: Important Points:
  • Practice questions covering a range of difficulty levels.
  • Reinforcing the understanding of integration concepts through application.

CBSE Class 12 Previous Year Question Papers

NCERT Solutions for Class 12 Maths Chapter 7

NCERT Solutions for Class 12 Maths Chapter 7 Integrals is very helpful for CBSE class 12 students. Find the detailed exercise-solutions to chapter 7 in the table below:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercises
Exercise 7.1
Exercise 7.2
Exercise 7.3
Exercise 7.4
Exercise 7.5
Exercise 7.6
Exercise 7.7
Exercise 7.8
Exercise 7.9
Miscellaneous Exercise

Benefits NCERT Solutions for Class 12 Maths Chapter 7

Here's a detailed discussion on the benefits of NCERT Solutions for Class 12 Maths Chapter 7: Conceptual Clarity: NCERT Solutions for Class 12 Maths Chapter 7 provide detailed explanations and step-by-step solutions for each exercise and example. This aids in enhancing conceptual clarity, ensuring that students understand the fundamental principles of integrals. Methodical Problem Solving: By offering systematic approaches and methodologies for solving problems, NCERT Solutions for Class 12 Maths Chapter 7 guide students through the process of integrating functions. This helps in developing a structured problem-solving mindset Comprehensive Coverage: The solutions cover all the topics within Chapter 7, from the introduction to integration as an inverse process of differentiation to the fundamental theorem of calculus and various techniques for evaluating integrals. This comprehensive coverage ensures that students are well-prepared for exams. Application of Concepts: NCERT Solutions for Class 12 Maths Chapter 7 provide numerous solved examples that illustrate the application of integration concepts to real-world problems. This helps students understand how to use integrals in different scenarios. Clarity on Standard Integrals: The solutions include discussions on standard integrals, helping students memorise and understand common integration formulas. This knowledge is crucial for efficient problem-solving. Visual Representation: NCERT Solutions for Class 12 Maths Chapter 7 often include visual representations of integrals, aiding in the understanding of geometric interpretations. This visual approach helps students connect mathematical concepts with their graphical representations. Preparation for Board Exams: Since NCERT is the recommended curriculum for CBSE board exams, using NCERT Solutions for Class 12 Maths Chapter 7 ensures that students are well-prepared for their exams. The solutions follow the same pattern as NCERT textbooks, making them an excellent resource for exam preparation. Self-Assessment and Practice: The exercise questions and additional problems provided in NCERT Solutions for Class 12 Maths Chapter 7 allow students to practise and assess their understanding. Regular practice using these solutions enhances problem-solving skills and builds confidence. Time Management: By providing clear and concise solutions, NCERT Solutions help students manage their time efficiently during exams. Understanding the methods and techniques reduces the time spent on solving problems. Free Accessibility: NCERT Solutions for Class 12 Maths Chapter 7 are often freely accessible online. This makes them easily available to all students, allowing them to supplement their learning without additional costs. Builds Strong Foundation for Further Studies: NCERT Solutions for Class 12 Maths Chapter 7 play a crucial role in building a strong foundation for further studies in calculus and mathematics. The concepts covered, including integration techniques and applications, serve as a stepping stone for advanced mathematical topics at the university level.

How to Prepare With NCERT Solutions for Class 12 Maths Chapter 7

Preparing for Class 12 Maths Chapter 7, which covers Integrals, with the help of NCERT Solutions can be a strategic and effective approach. Here's a detailed guide on how to prepare using NCERT Solutions:
  1. Understand the Basics: Start by thoroughly understanding the basic concepts introduced in Chapter 7. Ensure that you have a clear understanding of integration as the inverse process of differentiation and the geometrical interpretation of definite integrals.
  1. Read NCERT Textbook: Begin by reading the NCERT Class 12 Maths textbook for Chapter 7. Understand the theory, examples, and exercises provided in the textbook. This forms the foundation for your preparation.
  1. Use NCERT Solutions for Examples: For each example in the NCERT textbook, refer to the corresponding NCERT Solutions. Analyse the step-by-step solutions provided. Focus on the reasoning behind each step to enhance your problem-solving skills.
  1. Solve Exercise Questions: Work through the exercise questions in the NCERT textbook on your own before referring to the solutions. This will help you identify areas where you need more practice and clarify doubts.
  2. Refer to NCERT Solutions for Practice: After attempting the exercise questions, refer to the NCERT Solutions. Compare your solutions with the provided solutions to identify mistakes and learn alternate methods of solving problems.
  3. Focus on Standard Integrals: Pay special attention to the standard integrals discussed in the chapter. Memorise these formulas as they are essential for solving integration problems efficiently. Use flashcards or regular practice to reinforce these formulas.
  4. Practice Regularly: Integration is a skill that improves with practice. Regularly solve problems from the NCERT textbook and additional exercises provided in NCERT Solutions. This repetitive practice will build confidence and proficiency in integration techniques.
  5. Use Visual Aids: Take advantage of any visual aids provided in the NCERT Solutions. Visualise the geometric interpretations of integrals to strengthen your understanding of the subject.
  6. Apply Concepts to Real-World Problems: Relate integration concepts to real-world scenarios. Understand how integrals can be used to find quantities such as area, displacement, or accumulated changes. This application-oriented approach will deepen your understanding.
  7. Seek Clarification for Doubts: If you encounter doubts or difficulties, don't hesitate to seek clarification from your teacher, classmates, or online resources. A clear understanding of the basics is crucial for mastering integration.
  8. Create a Study Schedule: Plan your study schedule, allocating specific times for reading the textbook, solving problems, and reviewing NCERT Solutions. Breaking down your study sessions can make the preparation process more manageable.
  9. Utilise Additional Resources: While NCERT Solutions are excellent, consider exploring additional resources, such as reference books, online tutorials, or practice tests, to broaden your understanding and exposure to different problem-solving techniques.
  10. Review Regularly: Periodically review the entire chapter, focusing on challenging topics or problems. This revision process ensures that you retain the knowledge and are well-prepared for exams.
  11. Simulate Exam Conditions: Closer to exams, simulate exam conditions by solving practice papers and previous years' question papers within a specified time limit. This helps improve time management and builds exam confidence.

NCERT Solutions for Class 12 Maths Chapter 7 FAQs

What is the fundamental difference between definite and indefinite integrals?

The main difference lies in their purpose. A definite integral is used to find the accumulated quantity or net change between two specified points, often representing area under a curve. On the other hand, an indefinite integral represents a family of functions, and its main use is to find antiderivatives.

How can I effectively use substitution in evaluating definite integrals?

Substitution involves replacing variables to simplify integrals. To use it effectively, choose a substitution that simplifies the integral and helps in evaluating it. Ensure that you adjust the limits of integration accordingly when working with definite integrals.

What are the standard integrals, and why are they important in integration?

Standard integrals are common integrals of elementary functions. Examples include the integral of x^n, e^x, sin x, cos x, etc. Memorising these standard integrals is crucial as they serve as building blocks for solving more complex integrals efficiently.

How does the Fundamental Theorem of Calculus connect antiderivatives and definite integrals?

The Fundamental Theorem of Calculus establishes a direct connection between antiderivatives and definite integrals. It states that if F(x) is an antiderivative of f(x), then the definite integral of f(x) from a to b is equal to F(b) - F(a). This theorem simplifies the process of evaluating definite integrals.

Can you provide tips for approaching integration problems involving products of functions?

Integration by parts is a technique used for problems involving the product of two functions. Remember the formula: ∫ u dv = uv - ∫ v du. When choosing u and dv, it's often helpful to prioritise functions that become simpler or have known integrals when differentiated or integrated.
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