NCERT Solutions for Class 12 Maths Exercise 7 Integrals Miscellaneous Exercise

NCERT Solutions for Class 12 Maths Chapter 7 Miscellaneous Exercise: Integrals is one of the most prominent topics of CBSE Class 12 Maths. Practising all these questions helps students easily tackle the question paper during final exams. To understand NCERT Solutions for Integrals for Class 12 Maths, students must solve the Miscellaneous Exercise using the integration technique by parts.

Click on the topic-wise links provided in the table below to get access to the NCERT Mathematics Chapter 7 chapter-wise important questions:

Embibe provides solutions to help students prepare for their Class 12 board exams. Many questions and books are available at Embibe for practice at the Embibe Learning Platform. The students must follow the Class 12 Maths Chapter 7 Miscellaneous Exercise Solutions to ensure they can revise the entire chapter. Keep scrolling for practice and gain full knowledge about the topic.

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NCERT Solutions for Class 12 Maths Chapter 7 MiscellaneousExercise: Important Topics

Students preparing for Class 12 exams can refer to Embibe’s solutions for Miscellaneous Exercises. These exercises are structured to help the students get a complete idea of Chapter 7 questions. The solution of Miscellaneous Exercise is also one of the best ways to speed up the preparation. Since these Miscellaneous Exercises cover questions from the entire chapter, they will come in handy for the last-minute preparations.

Furthermore, these solutions follow the latest marking scheme that makes it easier for the students to understand the exam pattern and the type of questions that they will need to encounter during their board exams. All the important topics to cover in the NCERT Solutions of Class 12 Maths Chapter 7 miscellaneous is provided in Embibe. Students can watch the videos and prepare for their exams accordingly.

NCERT Solutions Class 12 Maths Chapter 7 Miscellaneous: Important Points to Remember

Given below are some of the important points of Maths chapter 7 that the students must follow while solving the NCERT solutions for Integrals:

Integration is the inverse process of Differentiation. In the differential calculus, we are given a function, and we have to find the derivative or differential of this function, but in the integral calculus, we are to find a function whose differential is given. Thus, integration is a process which is the inverse of Differentiation.

From the geometric point of view, an indefinite integral is a collection of a family of curves obtained by translating one of the curves parallel to itself upwards or downwards along the y-axis.

Integration by substitution: A change in the variable of integration often reduces an integral to one of the fundamental integrals. The method in which we change the variable to another variable is called the substitution method. When the integrand involves some trigonometric functions, we use some well-known identities to find the integrals.

Integration by parts: The integral of the product of two functions = first function × integral of the second function – integral of {differential coefficient of the first function × integral of the second function}. Care must be taken in choosing the first function and the second function. We must take that function as the second function whose integral is well-known to us.

We have defined ( ) b an f x dx ∫ as the area of the region bounded by the curve y = f (x), a ≤ x ≤ b, the x-axis and the ordinates x = a and x = b. Let x be a given point in [a, b]. Then ( ) x a f x dx ∫ represents the area function A (x). This concept of area function leads to the Fundamental Theorems of Integral Calculus.

First fundamental theorem of integral calculus: Let the area function be defined by A(x) = ( ) x a f x dx ∫ for all x ≥ a, where the function f is assumed to be continuous on [a, b]. Then A′(x) = f (x) for all x ∈ [a, b].

Second fundamental theorem of integral calculus: Let f be a continuous function of x defined on the closed interval [a, b] and let F be another function such that F( ) ( ) d x f x dx = for all x in the domain of f, then ( ) F( ) C F ( ) F ( ) [ ] b b a a f x dx x b a = + = − ∫ . It is called the definite integral of f over the range [a, b], where a and b are called the limits of integration, a being the lower limit and b the upper limit.

NCERT Solutions for Class 12 Maths: All Chapters

Students can go through the description of each chapter to get an overview of the contents of the chapters:

Steps to Practice Unlimited 12th NCERT Mathematics Chapter-wise Mock Tests

Embibe provides personalised mock tests for every student. They can practice in their own pace and prepare themselves with ease. Below we have provided a detailed guide to taking the mock test based on latest syllabus on Embibe:

1st Step: Click on ‘Create Account‘. 2nd Step: Enter your mobile number. 3rd Step: Then select your Goal as ‘CBSE’. 4th Step: Select your Exam Name as ’12th CBSE’. 5th Step: Click Next and select your preferred language between ‘English’ and ‘Hindi’. 6th Step: Once done, click on the below-mentioned links to take the TEST.

FAQs on NCERT Solutions for Class 12 Maths Chapter 7 Miscellaneous Exercises

Some of the frequently asked questions on NCERT Solutions for Class 12 Maths Chapter 7 Misc exercises is as follows:

Q: Can Integrals be helpful in preparing for competitive exams?

Ans: Yes, Integrals can be very helpful in preparing for competitive exams.

Q: Are the NCERT Solutions for Class 12 available for?

Ans. Yes. Students can refer to the Embibe Learning Platform to get solutions.

Q: Why is it important to solve Miscellaneous Exercise?

Ans: Miscellaneous Exercises cover questions from the entire chapter. They will come in handy for last-minute preparations.

Q: How to score well in Mathematics?

Ans: Practice with Embibe to score good marks in Maths Exam.

Q: What are the benefits of NCERT Solutions for Class 12?

Ans: As top teachers and Subject Matter Experts curate it, the concepts are perfectly explained.

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