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RD Sharma Class 12 Maths Solutions Chapter 20 - Definite Integrals

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Last updated date: 29th Nov 2023
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MVSAT Dec 2023

RD Sharma Solutions for Class 12 Maths Chapter 20 - Definite Integrals - Free PDF Download

In the provided pdf, we are going to learn about the chapter’s definite integrals. The definite integral of a function is related to the antiderivative and indefinite integral of a function. This chapter contains lots of formulae related to change of limits etc. and hence demands consistent practice. It is an important topic of the IIT JEE Mathematics syllabus and so students are advised to learn this concept.

Competitive Exams after 12th Science

Class 12 RD Sharma Textbook Solutions Chapter 20 - Definite Integrals

A brief overview of RD Sharma class 12 solutions definite integrals:

  • Definite integrals as a limit of the sum.

  • The first fundamental theorem of integral Calculus.

  • The second fundamental theorem of integral Calculus.

  • Basic properties of definite integrals.

  • Evaluation of definite integrals.


Benefits of RD Sharma Solutions for Class 12 Maths Chapter 20 - Definite Integrals

There are innumerous benefits of solving RD Sharma Solutions for Class 12 Maths Chapter 20 - Definite Integrals. A subject like mathematics requires adequate question practice in order to build strong concepts. While NCERT is a non-negotiable to score well in board exams or any competitive examination for that matter, students often look for other books to solve once they have mastered NCERT to improve their scores.


One such best book is RD Sharma for Class 12 Maths. Very often, teachers recommend it to students. Many toppers solve it too! 


While solving RD Sharma Class 12 Maths Chapter 20 - Definite Integrals, students might face problems in some questions. To help students with this, we have provided RD Sharma Solutions for Class 12 Maths Chapter 20 - Definite Integrals. Our expert teachers here at Vedantu have carefully formulated RD Sharma Solutions for Class 12 Maths Chapter 20 - Definite Integrals keeping in mind the problems students may face while solving questions. The teachers are the best in their field and are aware of the doubts students face while solving questions. The RD Sharma Solutions for Class 12 Maths Chapter 20 - Definite Integrals are highly detailed and easy to understand. 


The RD Sharma Solutions for Class 12 Maths Chapter 20 - Definite Integrals here are free to download. 


Students can find the RD Sharma Solutions for Class 12 Maths for all the chapters for free to download on this website.


Going through the RD Sharma Solutions for Class 12 Maths Chapter 20 - Definite Integrals will help students immensely and make them more satisfied with their preparation.


RD Sharma Solutions For Class 12 Maths Chapter 20 - Free PDF Download

We have provided step-by-step solutions for all exercise questions given in the pdf of Class 12 RD Sharma Chapter 20 - Definite Integrals. All the Exercise questions with solutions in Chapter 20 - Definite Integrals are given below:

Exercise 20.1

Exercise 20.2

Exercise 20.3

Exercise 20.4

Exercise 20.5


The summary of the topics listed above is given below:

Let x be a given point in [a, b], then \[\int_{a}^{b} f(x)dx\] is defined as the area of the region bounded by the curve y = f (x),  the x-axis and the ordinates x = a,  x = b.

 

The First Fundamental Theorem of Integral Calculus:

Let f(x) be a continuous function in the closed interval [a, b],  and if the area function is defined by A(x) = \[\int_{a}^{x} f(x)dx\] for all  x ≥ a. Then A′(x) = f (x) for all x ∈ [a, b].

 

The 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 d/dx(F(x)) = f(x) for all x in the domain of f, then \[\int_{a}^{b}f(x)dx = [F(x) + C]_{a}^{b} = F(b) - F(a)\].

 

This is called the definite integral of f over the range [a, b], and ‘a’ and ‘b’ are known as the limits of integration. Here  ‘a’, ‘b’ is called lower limit and upper limit respectively.

 

Properties of Definite Integrals

  • \[\int_{a}^{b} 1dx = b - a\]

  • \[\int_{a}^{a} f(x)dx = 0\]

  • \[\int_{a}^{b} f(x)dx  = \int_{a}^{b} f(t)dt\]

  • \[\int_{a}^{b} f(x)dx =  -\int_{b}^{a} f(x)dx\]

  • \[\int_{a}^{b} f(x)dx = \int_{a}^{c} f(x)dx + \int_{c}^{b} f(x)dx\]

  • \[\int_{a}^{b} f(x)dx = \int_{a}^{b} f(a + b - x)dx\]

  • \[\int_{-a}^{a} f(x)dx = 0\] (only if f(x) is odd)

  • \[\int_{-a}^{a} f(x)dx = 2\int_{0}^{a} f(x)dx\] (only if f(x) is even)


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Conclusion

Definite Integral is one of the most important chapters in terms of the exam. The RD Sharma book is very useful and contains a detailed explanation of every topic in this chapter. The examples and practice problems for each topic are given separately. Students can download Rd Sharma class 12 solutions definite integrals from the link given above. RD Sharma solutions provided by subject experts of Vedantu are very clear and easy to understand. The formula used in the steps of any problem is clearly mentioned in the solution, thus it becomes easy for the students to practice enough questions from this chapter and prepare well.

FAQs on RD Sharma Class 12 Maths Solutions Chapter 20 - Definite Integrals

1. What is the Geometric Interpretation of Definite Integrals?

The geometric interpretation of the definite integral is related to the area. The definite integral ∫ab f(x)dx is the area of the region bounded by the curve y = f (x),  the x-axis and the ordinates x = a and x = b.

2. Can the Upper Limit be Less than the Lower Limit of Integration?

There is no such restriction. The upper limit of the integration can be smaller than the lower limit of the integration. Collectively, we can call the upper and lower limit the interval of integration.

3. What Happens to the Definite Integral When we Interchange the Upper and Lower Limits of Definite Integral?

Let F(x) be the antiderivative of f(x), then 

ab f(x)dx =  F(b) - F(a)

and ∫ab f(x)dx =  F(a) - F(b) = - ( F(b) - F(a) ) = - ∫ab f(x)dx

Therefore on interchanging the limits of the integration the magnitude of result remains the same, but the sign is reversed. For example

12 x2dx  = [x3/3]12 = (4 - 1)/3 = 1

and  ∫12 x2dx  = [x3/3]21 = (1 - 4)/3 = -1

4. How to download RD Sharma Solutions for Class 12 Maths Chapter 20 - Definite Integrals?

It is very simple to download RD Sharma Solutions for Class 12 Maths Chapter 20 - Definite Integrals PDF for free from this website. You just have to sign up to the website and click on the download button. That’s it! This will get the RD Sharma Solutions for Class 12 Maths Chapter 20 - Definite Integrals downloaded for free on your device. All the solutions on this website are free of cost. You don’t have to pay anything for it.

5. How to prepare RD Sharma for Class 12 Maths Chapter 20 - Definite Integrals?

To study RD Sharma Class 12 Maths Chapter 20 - Definite Integrals, first read the text given in the book. Then solve the examples. Once you are done with the examples, solve the exercises. If you get stuck while solving RD Sharma Class 12 Maths Chapter 20 - Definite Integrals questions, try them again. If you are still unable to figure it out, you can take the help of RD Sharma Solutions for Class 12 Maths Chapter 20 - Definite Integrals uploaded on our website.

6. How to clear our doubts on RD Sharma Solutions for Class 12 Maths Chapter 20 - Definite Integrals?

It is very important for students to get their doubts cleared. Most of the time students are too scared to ask their doubts to teachers. They avoid asking doubts and leave their concepts unclear. We highly encourage you to get your doubts cleared. If you are too shy to ask your teacher, you can make use of the free solutions provided here. If you still have any doubts, you can ask our experts but make sure to understand your doubts.