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NCERT Solutions for Class 12 Maths Chapter 7 – Integrals

NCERT Solutions for Class 12 Maths Chapter 7- Integrals have been carefully prepared on the basis of the latest CBSE Class 12 Maths Syllabus. This Chapter is one of the most important Chapters of the Class 12 Maths curriculum. It covers various significant concepts such as indefinite integrals, properties of indefinite integrals, definite integrals, and properties of definite integrals.

This Chapter focuses deeply on various methods of integration, such as integration using trigonometric identities, integration by substitution, integration by parts, and integration by partial fractions. This Chapter also teaches you the first fundamental theorem of calculus and the second fundamental theorem of calculus. Our NCERT Solutions cover all concepts of this Chapter in details and in a step by step manner.

This Chapter comprises 11 comprehensive exercises and a miscellaneous exercise that gives you good enough practice to understand and become thorough with every concept of this Chapter. Our NCERT Solutions answer all questions of all exercises accurately, precisely and in an easy to understand manner to help you become conceptually sound and score high marks in your Class 12 Board Exams. They have been designed in a manner to make learning Maths fun, interesting, stimulating, enjoyable and highly effective. With our NCERT Solutions at your disposal, you can now learn from anywhere and anytime at your speed and convenience.

NCERT CBSE Class 12 Maths Chapter 7: Important Topics

NCERT CBSE Class 12 Maths Chapter 7 is very important from Class 12 Board exam point of view. Some of the significant topics covered in this Chapter have been discussed below.

  • Indefinite Integrals

In this section of CBSE NCERT Class 12 Maths Book, you will learn about the definition of the indefinite integrals. In this, you will learn how indefinite integral is the compilation of the curves if seen from the geometrical perspective. It is an integral that does not have any lower or upper limit.

  • Properties of Indefinite Integrals

In this topic, NCERT Solutions for Class 12 Maths Chapter 7- Integrals, you will understand several properties of Indefinite Integrals. You will learn properties like how to split two functions when you have to add those and find the integral. You will also learn How to bring constants out of the integral. 

Some of the properties are cites below

  1. The indefinite integral of the sum of two functions is equal to the sum of the integrals:
    ∫[f(x)+g(x)]dx= ∫f(x)dx+∫g(x)dx.
  2. The indefinite integral of a function f(x) is the collection of all antiderivatives for this function:
    ∫f(x)dx=F(x)+C, if F′(x)=f(x).
  3. The indefinite integral of the difference of two functions is equal to the difference of the integrals:
    ∫[f(x)−g(x)]dx= ∫f(x)dx−∫g(x)dx.
  4. A constant factor can be moved across the integral sign:
    ∫kf(x)dx=k∫f(x)dx.

 

  • Methods of Integration

 

In this section of Class 12, Maths Integrals covers various methods of integration such as the following.

 

  • Integration using trigonometric identities

 

When you see that the integrand involves certain trigonometric functions, then you must use the following identities to solve the integral.

  1. 2 sin A cos B = sin(A + B) + sin(A − B) 
  2. 2 cos A cos B = cos(A − B) + cos(A + B) 
  3. 2 sin A sin B = cos(A − B) − cos(A + B)
  4. sin2 A + cos2 A = 1
  5. sin 2A = 2 sin A cos A
  6. 1 + tan2 A = sec2 A

(some of the commonly used trigonometric identities)

 

  • Integration by substitution

 

Integration by substitution is carried out when a suitable substitution of variable results in simplification of the integral. Our NCERT Solutions explain this concept in a step by step manner with the help of solved examples.

 

  • Integration by parts

 

Another important method of integration covered in NCERT Class 12 Maths Chapter 7 is the integration by parts. This method is useful when two functions are multiplied together. The formula of the same has been given below.

∫u v dx = u∫v dx −∫u’ (∫v dx) dx

Some other significant topics covered in this NCERT Solutions for Class 12 Maths Chapter 7-Integrals include definite integrals, properties of definite integrals, the first fundamental theorem of calculus, the second fundamental theorem of calculus, and integration by partial fractions. 

NCERT Class 12 Maths Chapter 7 Exercises

NCERT Class 12 Maths Chapter 7 Integrals is an important module in Class 12 Maths CBSE syllabus. Many questions in CBSE Board exams are asked from this Chapter. In this Chapter, there are a total of 11 exercises. They have been discussed below.

Exercise 7.1

Exercise 7.1 of this Chapter has 22 questions. Questions 1 to 5 asks you to find an antiderivative of the given functions by the method of inspection. Questions 6 to 20 ask you to find the given integrals. Questions 21 and 22 are MCQs.

Exercise 7.2

Exercise 7.2 has 39 questions. Questions 1 to 37 asks you to integrate the given functions. Questions 38 and 39 are MCQs.

Exercise 7.3

Exercise 7.3 comprises 24 questions. They deal with the advanced concepts which do require a method of substitution and various other trigonometric identities. Question 23 and 24 are MCQs.

Exercise 7.4

Exercise 7.4 has 25 questions. Questions 1 to 23 needs you to integrate the given functions. Questions 24 and 25 are MCQs. This exercise covers various methods of integration, such as integration by substitution.

Exercise 7.5

Exercise 7.5 consists of 23 questions. Questions 1 to 21 asks you to integrate rational functions. Questions 22 and 23 are MCQs. Many questions from this exercise deal with the concept of integration by parts.

Exercise 7.6

Exercise 7.6 has 24 questions. Questions 1 to 22 asks you to integrate the given functions. Questions 23 and 2 are MCQs. Many questions from this exercise deal with the concept of integration by partial fractions.

Exercise 7.7

In this exercise, you need to find the unique value of integral functions in all the 11 questions by applying the limits in each question. Questions 10 and 11 are MCQs.

Exercise 7.8

Exercise 7.8 has 6 questions that need you to evaluate the given definite integrals as limits of sums.

Exercise 7.9

In this section, you need to solve questions of definite integrals using the substitution method. All 22 questions need to be solved with a similar approach. However, questions 21 and 22 are MCQs.

Exercise 7.10

It deals with several new concepts of indefinite integrals and covers 10 questions. Questions 9 and 10 are MCQs. Our NCERT solutions will help you to build an in-depth understanding of calculus.

Exercise 7.11

Exercise 7.11 has 21 questions. Questions 1 to 19 needs you to use the properties of definite integrals and evaluate the given integrals. Questions 20 and 21 are MCQs.

NCERT CBSE Chapter 7 Integrals contains a miscellaneous exercise which contains questions from all the different concepts of the Chapter. aIt comprises a total of 44 questions.

Benefits of Class 12 Maths Integrals Solutions

There are a plethora of benefits associated with learning and practising from our smart, systematic and logical NCERT Solutions for Class 12 Maths Chapter 7-Integrals. Some of them have been cited below.

  • Our NCERT Solutions give you ample practice to help you become thorough with all topics and sub-topics of the Chapter quickly and effectively.
  • They answer all questions from all exercises in a step by step manner and in a simple language.
  • They cover a wide range of question types important from Class 12 Board Exam point of view.
  • Our NCERT Solutions have been prepared on the basis of the latest guidelines given by NCERT and CBSE. 
  • Our smart study techniques and learning shortcuts help you learn quickly and score more marks.
  • They have been designed to clear all your doubts and improve your retention rate.
  • They have been solved using the best methods of solving the problems.