NCERT Solutions for Class 12 Mathematics Chapter 7 Integrals introduces ‘Integration,’ the other side of differentiation. Differentiation provides us with the rate of change, or in geometric terms, the slope of the tangent, at every given point in the curve, while Integration gives us the area under the curve. Integrals are the inverse of differentiation; therefore, we can get the original function back by integrating the resultant derived from differentiating a function. NCERT solutions Class 12 Mathematics Chapter 7 explains what Integration is and how to use it, as well as the numerous methods for calculating it.
The integration method allows you to sum infinitesimally small parts infinitely many times, allowing you to calculate the area under the curve. With the help of examples and easily comprehensible methods provided by Extramarks, students find the concept intuitive and are able to compute integration quickly. Chapter 7 Mathematics Class 12 Integrals teaches students about integration, Its types (indefinite and definite) as well as their relationship to differentiation. It also outlines multiple approaches that can be used to achieve integration. Integration is a useful method for resolving scientific and technical issues. It can also be used to answer questions in economics, finance, and probability.
Key Topics Covered in NCERT Solutions Class 12 Mathematics Chapter 7
Integration, a process opposite of differentiation, consists of functions known as integrals that must satisfy a given differential equation. Students shall be responsible for identifying the functions whose differential will be given to them. Here are some concepts that will be covered in this paper.
- Integration
- Indefinite integrals
- Application of integrals
- Integrals for class 12
- Application of integrals for class 12
- Integral calculus
Indefinite integrals, geometrical interpretation of indefinite integrals, properties of indefinite integrals, standard integrals, and methods of integration including integration by substitution method, integration by partial fractions, and integration by parts are among the significant topics covered in NCERT Solutions Class 12 Mathematics Chapter 7. Definite integrals, the fundamental theorem of calculus, and the properties of definite integrals are some of the other essential topics covered in this chapter as well. The goal of curating these solutions is to promote fundamental knowledge of integral calculus to assist students in their studies.
List of NCERT Solutions Class 12 Mathematics Chapter 7 Exercises & Answer Solutions
One of the most important topics in calculus – integration, has a wide range of practical applications. This lesson contains several formulae and hence necessitates laser-sharp focus. Revising the solutions regularly is the greatest approach to remember these concepts and measure your understanding of integration. The following is an exercise-by-exercise detailed analysis of NCERT Solutions Class 12 Mathematics Chapter 7 Integrals to assist students in developing a strong understanding of this subject:
Chapter 7 Ex 7.1 Solutions – 22 Questions – Class 12 Mathematics
Chapter 7 Ex 7.2 Solutions – 39 Questions – Class 12 Mathematics
Chapter 7 Ex 7.3 Solutions – 24 Questions – Class 12 Mathematics
Chapter 7 Ex 7.4 Solutions – 25 Questions – Class 12 Mathematics
Chapter 7 Ex 7.5 Solutions – 23 Questions – Class 12 Mathematics
Chapter 7 Ex 7.6 Solutions – 24 Questions – Class 12 Mathematics
Chapter 7 Ex 7.7 Solutions – 11 Questions – Class 12 Mathematics
Chapter 7 Ex 7.8 Solutions – 6 Questions – Class 12 Mathematics
Chapter 7 Ex 7.9 Solutions – 22 Questions – Class 12 Mathematics
Chapter 7 Ex 7.10 Solutions – 10 Questions – Class 12 Mathematics
Chapter 7 Ex 7.11 Solutions – 21 Questions – Class 12 Mathematics
Chapter 7 Miscellaneous Exercise – 44 Questions – Class 12 Mathematics
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NCERT Solutions Class 12 Mathematics Chapter 7 Formula List
NCERT solutions Class 12 Mathematics chapter 7 includes various important concepts necessary for mastering higher-level mathematics. Students must understand the derivation of the principles and formulas given. This will assist them in implementing the measures necessary to solve integration problems and limit their mistakes. Students should also keep a formula chart to quickly and efficiently review the formulae before exams. The following are some key formulae mentioned in NCERT Solutions for Class 12 Mathematics Chapter 7:
- ∫ f(x) dx = F(x) + C
- Power Rule: ∫ xn dx = (xn+1)/ (n+1)+ C. (Where n ≠ -1)
- Exponential Rules: ∫ ex dx = ex + C
- ∫ ax dx = ax /ln(a) + C
- ∫ ln(x) dx = x ln(x) -x + C
- Constant Multiplication Rule: ∫ a dx = ax + C, where a is the constant.
- Reciprocal Rule: ∫ (1/x) dx = ln(x)+ C
- Sum and Difference Rules:
- ∫ [f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx
- ∫ [f(x) – g(x)] dx = ∫f(x) dx – ∫g(x) dx
- ∫ k f(x) dx = k ∫f(x) dx, where k is any real number.
- Integration by parts: ∫ f(x) g(x) dx = f(x) ∫ g(x) dx – ∫[d/dx f(x) * ∫ g(x) dx]dx
- ∫ cos x dx = sin x + C
- ∫ sin x dx = -cos x + C
- ∫ sec 2x dx = tan x + C
- ∫ cosec 2x dx = -cot x + C
- ∫ sec x tan x dx = sec x + C
- ∫ cosec x cot x dx = – cosec x + C
NCERT Class 12 Mathematics Syllabus CBSE
Term – 1
| Unit Name
|
Chapter Name |
|
Relations and Function |
Relations and Functions
Inverse Trigonometric Functions |
| Algebra
|
Matrices
Determinants |
| Calculus
|
Continuity and Differentiability
Application of Derivatives |
| Linear Programming |
Linear Programming |
Term – 2
| Unit Name |
Chapter Name |
|
Calculus
|
Integrals
Application of Integrals
Differential Equations |
| Vectors and Three-Dimensional Geometry |
Vector Algebra
Three Dimensional Geometry |
| Probability |
Probability |
NCERT Class 12 Mathematics Exam Pattern
| Duration of Marks |
3 hours 15 minutes |
| Marks for Internal |
20 marks |
| Marks for Theory |
80 marks |
| Total Number of Questions |
38 Questions |
| Very short answer question |
20 Questions |
| Short answer questions |
7 Questions |
| Long Answer Questions (4 marks each) |
7 Questions |
| Long Answer Questions (6 marks each) |
4 Questions |
Key features NCERT Mathematics class 12th chapter 7
Students can study the following topics by learning the NCERT Solutions for Integrals:
As the inverse of differentiation, integration Substitution, partial fractions, and parts are used to integrate a number of functions. Simple integrals of the following categories and problems dependent on them are evaluated. Fundamental Theorem of Calculus, Definite Integrals as a Limit of a Sum (without proof). Evaluation of definite integrals and basic properties of definite integrals.
NCERT Exemplar Class 12 Mathematics
All solutions and problems are given to help students prepare for their final exams. These example questions are a little more complex, and they cover each and every concept covered in each chapter of the Class 12 Mathematics subject. Students will fully understand all the concepts covered in each chapter by practising these NCERT exemplar for Mathematics Class 12. Exemplars provide the best solutions to challenges that students confront. To match the ideas taught in each class and provide the greatest practising materials or worksheets for students, all of these questions strictly follow updated 2022-23 CBSE guidelines.