NCERT Solutions Class 12 Maths Chapter 7 Integrals
NCERT Solutions for Class 12 Maths Chapter 7 Integrals is an introduction to the other side of differentiation that is ‘Integration’. Differentiation gives us the rate of change or in terms of geometry the slope of the tangent at any given point in the curve however, it is by using Integration that we get to know the area under the curve. Integrals are inverse of differentiation, so if we differentiate a function, then the integration of the resultant gives us back the original function. NCERT solutions Class 12 maths Chapter 7 exposes the meaning and implementation of integration and provides different ways of calculating it.
Class 12 Maths Chapter 7 Integrals helps kids understand Integration, its type that is indefinite and definite integral, its relation with differentiation and the different ways integration can be done. Integration is a practical tool that is used to solve scientific and engineering problems. It helps to solve various questions from economics, finance, as well as probability. The pdf file for Class 12 maths NCERT solutions Chapter 7 Integrals can be accessed through the links given below:
- NCERT Solutions Class 12 Maths Chapter 7 Ex 7.1
- NCERT Solutions Class 12 Maths Chapter 7 Ex 7.2
- NCERT Solutions Class 12 Maths Chapter 7 Ex 7.3
- NCERT Solutions Class 12 Maths Chapter 7 Ex 7.4
- NCERT Solutions Class 12 Maths Chapter 7 Ex 7.5
- NCERT Solutions Class 12 Maths Chapter 7 Ex 7.6
- NCERT Solutions Class 12 Maths Chapter 7 Ex 7.7
- NCERT Solutions Class 12 Maths Chapter 7 Ex 7.8
- NCERT Solutions Class 12 Maths Chapter 7 Ex 7.9
- NCERT Solutions Class 12 Maths Chapter 7 Ex 7.10
- NCERT Solutions Class 12 Maths Chapter 7 Ex 7.11
- NCERT Solutions Class 12 Maths Chapter 7 Miscellaneous Exercise
NCERT Solutions for Class 12 Maths Chapter 7 PDF
The integration provides a way to sum infinitesimally small parts infinite times; this gives the ability to find the area under the curve. With the help of examples, students will eventually find the concept intuitive and using the methods provided, will be able to calculate integration easily. The pdf of the NCERT solutions Class 12 maths Chapter 7 can be found below:
☛ Download Class 12 Maths NCERT Solutions Chapter 7 Integrals
NCERT Class 12 Maths Chapter 7
NCERT Solutions for Class 12 Maths Chapter 7 Integrals
Integration is one of the most important topics in calculus that has wide-ranging practical implications. This lesson is laced with a number of formulas and hence, requires clear laser-sharp focus. The best way to memorize these formulas as well as gauge the understanding of integration is by diligently revising the solutions periodically. An exercise-wise detailed analysis of NCERT Solutions Class 12 Maths Chapter 7 Integrals is given below to help children instill a robust knowledge of this subject matter :
- Class 12 Maths Chapter 7 Ex 7.1 - 22 Questions
- Class 12 Maths Chapter 7 Ex 7.2 - 39 Questions
- Class 12 Maths Chapter 7 Ex 7.3 - 24 Questions
- Class 12 Maths Chapter 7 Ex 7.4 - 25 Questions
- Class 12 Maths Chapter 7 Ex 7.5 - 23 Questions
- Class 12 Maths Chapter 7 Ex 7.6 - 24 Questions
- Class 12 Maths Chapter 7 Ex 7.7 - 11 Questions
- Class 12 Maths Chapter 7 Ex 7.8 - 6 Questions
- Class 12 Maths Chapter 7 Ex 7.9 - 22 Questions
- Class 12 Maths Chapter 7 Ex 7.10 - 10 Questions
- Class 12 Maths Chapter 7 Ex 7.11 - 21 Questions
- Class 12 Maths Chapter 7 Miscellaneous Exercise - 44 Questions
☛ Download CBSE Class 12 Maths Chapter 7 NCERT Book
Topics Covered: The major sections in this chapter are the definition of Integration, relation with differentiation, integration methods, method of substitution, integration of particular functions, integration of partial functions as well as definite and indefinite integrals. Other topics included in these Class 12 maths NCERT solutions Chapter 7 are the first and the second Fundamental Theorem of integral calculus and how to do integration by parts.
Total Questions: Class 12 Maths Chapter 7 Integrals consists of a total of 227 questions of which 150 are easy, 70 are moderate and 7 are long answer-type. Apart from this, there are 44 miscellaneous problems spanning the entire integral topic.
List of Formulas in NCERT Solutions Class 12 Maths Chapter 7
NCERT solutions Class 12 maths Chapter 7 covers lots of important concepts crucial for understanding higher grade maths. Students must make it a point to understand the derivation of the rules and formulas included. This will help them in applying the steps required for integration problems and reduce the scope of making mistakes. Kids are also advised to keep a formula chart enabling them to revise the formulas quickly and efficiently before examinations. Some important formulas covered in NCERT Solutions for Class 12 maths Chapter 7 are given below:
- ∫ 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
- ∫ sec2 x dx = tan x + C
- ∫ cosec2 x dx = -cot x + C
- ∫ sec x tan x dx = sec x + C
- ∫ cosec x cot x dx = - cosec x + C
FAQs on NCERT Solutions for Class 12 Maths Chapter 7
Why are NCERT Solutions Class 12 Maths Chapter 7 Important?
NCERT Solutions for Class 12 Maths Chapter 7 are well-designed by subject experts after thorough research to impart a deep knowledge of all basics. Strategically preparing all the important topics with the help of these resources will enable students to attain a thorough conceptual understanding to perform well in exams. The well-crafted format of these solutions is excellent to encourage math practice which in turn helps in enhancing the confidence of students.
Do I Need to Practice all Questions Provided in NCERT Solutions Class 12 Maths Integrals?
Integrals is a vast topic that requires a lot of practice along with building the knowledge of newer concepts on previously learned topics. Practicing all the questions included in NCERT Solutions Class 12 Maths Integers will help students develop a gradual understanding of subsequent topics. All the sums incorporated in these solutions are carefully placed to offer step-wise learning for adjusting to the increasing complexity of the chapter. It will also enable kids to attain the right approach for tackling complicated problems confidently by enhancing their problem-solving skills.
What are the Important Topics Covered in Class 12 Maths NCERT Solutions Chapter 7?
The important topics covered in the NCERT Solutions Class 12 Maths Chapter 7 are indefinite integrals, geometrical interpretation of indefinite integral, properties of indefinite integrals, standard integrals, methods of integration including integration by substitution method, integration by partial fractions, and integration by part. Some of the other important topics included in this chapter are based on definite integrals, the fundamental theorem of calculus, and the properties of definite integrals. The objective of curating these solutions is to promote the fundamental knowledge of integral calculus to help students in their studies.
How Many Questions are there in NCERT Solutions for Class 12 Maths Chapter 7 Integrals?
There are a total of 227 questions provided in 11 exercises of NCERT Solutions Class 12 Maths Chapter 7 Integrals. Out of these 227 sums, there are 150 problems that are simple and easy to solve, whereas the rest of the 77 range between moderate to complex long answer questions that need more practice of fundamental concepts. Additionally, there is one miscellaneous exercise provided in this chapter with 44 questions to help the students get thorough with the concept of Integrals.
How CBSE Students can utilize NCERT Solutions Class 12 Maths Chapter 7 effectively?
To effectively utilize the NCERT Solutions Class 12 Maths Chapter 7 students should first have a strong foundation of all the core concepts and formulas of integral calculus. The second step is to establish a deep understanding of the link between differential and integral calculus. Finally, students can apply these topics to various levels of questions and use the solutions to check if their answers, as well as steps, are correct.
Why Should I Practice NCERT Solutions Class 12 Maths Integrals Chapter 7?
Implementing a constant practice of the NCERT Solutions Class 12 Maths Integrals Chapter 7 is a fast and steady method for students to build a strong mathematical foundation for solving sums related to integral calculus. These solutions will guide them the right way to crack complicated sums in both boards or competitive exams. Additionally, by revising these solutions kids can gain a clear conceptual fluency that will even help them in their careers.