RS Aggarwal Class 12 Chapter 12 Solutions (Indefinite Integral)
RS Aggarwal Solutions for Class 12 Chapter 12 are created with the viewpoint of bringing you at ease with the complex topics of the chapter. Indefinite Integral is a fundamental chapter that will help you understand the chapters ahead. In this chapter, you will be introduced to the concepts of anti-derivatives and area function. You will also get to understand the most important theorems of integral which are the first fundamental theorem of Calculus and the second fundamental theorem of Calculus that have a broad range of applications in maths and other subjects.
There is only one exercise that has 32 questions and an additional exercise of objective 41 questions covering a wide variety of functions under each topic. If you are able to grasp the methodology of how to solve these questions, you will be able to approach the questions in exams correctly and with confidence. These questions accompanied by your textbook questions are enough to prepare you for your school exam of CBSE, UP board etc. Solving these questions using Instasolv as your source of reference will make you adhere to the guidelines of the board.
The Maths experts of Instasolv have prepared these answers after thorough research and massive experience with students. Thus, they are well aware of your vulnerable point where you might end up losing marks. You will find the RS Aggarwal solutions accurate and detailed.
Summary of Important Topics Covered in RS Aggarwal Solutions for Class 12 Chapter 12 – Indefinite Integrals
Integration is a method of adding things up. It joins slices and makes them whole.
Integration is the inverse of Differentiation. To find the derivative of f(x), we use:
So to calculate the antiderivative, we use –
∫ g(x) dx = f(x) + c
Here, c is the constant of integration.
By definition, ∫ab f(x) dx is the area of the region bounded by the curve y = f(x), the x-axis and the coordinates ‘x = a’ and ‘x = b’. If ‘x’ is a point in [a, b], then ∫ax f(x) dx shows the area under the curve y.
First Fundamental Theorem of Calculus:
If ‘f’ is a continuous function on the closed interval [a, b] and A (x) is the area function. Then A′(x) = f (x), for all x ∈ [a, b].
Second Fundamental Theorem of Calculus:
If ‘f’ is a continuous function defined on the closed interval [a, b] and F is an anti-derivative of ‘f’. Then
∫ab f(x) dx = [F(x)]ab = F(b) – F(a)
The area under the curve:
For now, we wanted to calculate the area of the curve y=f(x), which will be
∫ f(x) dx = h(x) + c
Formulae for Indefinite Integrals:
Some of the basic indefinite integrals formulae are:
- xn dx = xn+1 + c
________ (n + 1)
- ∫ sin x dx = – cos x + c
- ∫ cos x dx = sin x + c
- ∫ 1 dx = ln x + c
- ∫ ex dx = ex + c
Properties of Indefinite Integrals:
- Theorem 1
The method of differentiation and integration are the opposite of each other.
Proof: If F be the anti-derivative of f, i.e., d⁄dx F(x) = f(x)
Then ∫ f(x) dx = F(x) + C
Hence, d⁄dx ∫ f(x) dx = d⁄dx [F(x) + C] = d ⁄ dx F(x) = f(x).
Similarly, f ′(x) = d ⁄ dx f(x) and hence ∫ f ′(x) dx = f(x) + C.
Where C is the constant of integration.
- Theorem 2
The integration of the sum of two integrands is the sum of integrations of two integrands.
∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
Proof: Using Theorem 1, we have
d⁄dx [∫ [f(x) + g(x)] dx] = f(x) + g(x) … (1)
d⁄dx [∫ f(x) dx + ∫ g(x) dx] = d⁄dx ∫ f(x) dx + d⁄dx ∫ g(x) dx = f(x) + g(x) … (2)
From (1) and (2), we have, ∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
- Theorem 3
Let k be any real number. We can say that, ∫ k f(x) dx = k ∫ f(x) dx.
Proof: Using theorem 1, d⁄dx ∫ k f(x) dx = k f(x) … (1)
Also, d⁄dx [∫ k f(x) dx] = k d⁄dx ∫ f(x) dx = k f(x) … (2)
From (1) and (2), we have, ∫ k f(x) dx = k ∫ f(x) dx.
The above result can be generalized to:
ʃ k1 [f1(x) + k2 f2(x) +…+ kn fn(x)] dx
= k1 ∫ f1(x) dx + k2 ∫ f2 (x) dx + … + kn ∫ fn(x) dx.
Exercise-Wise Discussion of Questions in RS Aggarwal Solutions for Class 12 Chapter 12 – Indefinite Integrals
- In the only exercise of RS Aggarwal Chapter 12, you will find a diverse set of questions covering all the topics in the chapter which makes these solutions crucial from the exam perspective.
- The questions are also of objective or short-answer type besides the classic long answers.
- These exercise solutions will give a very apt idea about the kind of questions that are asked in CBSE exams.
- The questions are arranged in an orderly manner so as to pace with your level of understanding as you advance towards more complex topics.
- It is advised that you do the final polishing of your concepts of Calculus before your exam using Instasolv for an interactive study session.
Benefits of RS Aggarwal Solutions for Class 12 Maths Chapter 12 by Instasolv
- The RS Aggarwal class 12 maths Solutions chapter 12 are prepared in easy to understand language by the experts at Instasolv.
- We, at Instasolv, believe that efficient utilisation of study hours is mandatory to stand out among the crowd, which you will witness once you will refer to our solutions. You can complete the whole chapter quickly without any distraction.
- The Instasolv experts have covered all the questions in detail without compromising the quality of the exams.