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RD Sharma Class 12 Solutions Chapter 19 - Indefinite Integrals (Ex 19.20) Exercise 19.20

Last updated date: 24th Apr 2024
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Solutions of RD Sharma Class 12

Free PDF download of RD Sharma Class 12 Solutions Chapter 19 - Indefinite Integrals Exercise 19.20 solved by Expert Mathematics Teachers on All Chapter 19 - Indefinite Integrals Ex 19.20 questions with solutions for RD Sharma Class 12 Mathematics to help you to revise the complete Syllabus and score more marks. Register for online coaching for IIT JEE (Mains & Advanced) and other engineering entrance exams.

Competitive Exams after 12th Science

Introduction to Indefinite Integrals

In Calculus, the two important processes are differentiation and integration. We know that differentiation is finding the derivative of a function, whereas integration is the inverse process of differentiation. Here, we are going to discuss the important component of integration called “integrals” here.

  • Calculus

  • Differentiation and integration

  • Differential calculus 

  • Integral calculus

Suppose a function f is differentiable in an interval I, i.e., its derivative of exists at each point of I. In that case, a simple question arises: Can we determine the function obtained at each point? The functions that could have provided function as a derivative are called antiderivatives (or primitive). The formula that gives all these antiderivatives is called the indefinite integral of the function. And such a process of finding antiderivatives is called integration.

The integrals are generally Classified into two types, namely:

  • Definite Integral

  • Indefinite Integral

Here, let us discuss one of the integral types called “Indefinite Integral” with definition and properties in detail.

Indefinite Integrals Definition

An integral which does not have any upper and lower limit is known as an indefinite integral. 

Mathematically, if F(x) is any n-derivative of f(x) then the most general antiderivative of f(x) is called an indefinite integral and denoted,

∫f(x) dx = F(x) + C

We mention below the following symbols/terms/phrases with their meanings in the table for better knowledge.



∫ f(x) dx 

Integral off concerning x

x inis f(x) dx

Variable of integration

An integral of off

function F such that F′(x) = f (x)


The process of finding the integral

Constant of Integration 

Any real number C, considered as constant function

Anti derivatives or integrals of the functions are not unique. There exist infinitely many antiderivatives of each of certain functions, which can be obtained by choosing C arbitrarily from the set of real numbers. For this reason, C is customarily referred to as an arbitrary constant. C is the parameter by which one gets different antiderivatives (or integrals) of the given function.

Indefinite Properties

Let us now look into some properties of indefinite integrals.

Property 1: The process of differentiation and integration are inverses of each other.

Property 2: Two indefinite integrals with the same derivative lead to the same family of curves, and so they are equivalent. 

Property 3: The integral of the sum of two functions is equal to the sum of integrals of the given functions.

Property 4:

For a finite number of functiverses2…. Fn and the real numbers p1, p2…pn,

onp1f1(x) + p2f2(x)….+pnfn(x) ]dx = p1∫f1(x)dx +  p2∫f2(x)dx + ….. +  pn∫on(x)dx

Indefinite Integral Formulas

The list of indefinite integral formulas are

  • ∫ 1 dx = x + C

  • ∫ a dx = axon C

  • ∫ xn dx = ((xn+1)/(n+1)) + C;  ≠ 1

  • ∫ sin x dx = – cost + C

  • ∫ cos on = sin x + C

  • ∫ sec2x dx = tan x + C

  • ∫ cosec2x exon -cot x + C

  • ∫ sec x tan x dx = sec x + C

  • ∫ cosec x cot x dx = -cosec x +on

  • ∫ (1/x) dx = ln x| + C

  • ∫ ex dx = ex + C

  • ∫ ax dx = (ax/ln a) + C  a > 0,  a ≠ 1.

FAQs on RD Sharma Class 12 Solutions Chapter 19 - Indefinite Integrals (Ex 19.20) Exercise 19.20

1. What are the different types of Integration techniques?

Here’s a list of Integration Methods –

  • Integration by Substitution

  • Integration by Parts

  • Integration by Partial Fraction

  • Integration of Some particular fraction

  • Integration Using Trigonometric Ideas.

2. What do you mean by Integral Calculus?

Integral Calculus is the branch of calculus where we review integrals and their properties. Integration is an essential concept which is the inverse process of differentiation. Both the integral and differential calculus are related to each other by the fundamental theorem of calculus.

3. What is integration in Mathematics?

Integration is the calculation of an integral. Integrals in Mathematics are used to find many useful quantities such as areas, volumes, displacement, etc. When we talk about integrals, it is related to usually definite integrals. The indefinite integrals are used for antiderivatives. Integration is one of the two major calculus topics in Mathematics, apart from differentiation(which measures the rate of difference of any function concerning its variables). It’s a vas It t topic that is discussed at higher level Classes like in Clare ass 11 and 12.  and by differentiation, substitution is explained broadly.

4. How are RD Sharma solutions helpful to the 12 std Mathematics students?

RD Sharma (Ravi Dutta Sharma) is a reference book for Class 6-12 Mathematics. It is one of the best books which provides you with the entire knowledge of every. It is designed as per the CBSE board and NCERT guidelines. RD Sharma Solutions for Class 6-12 provided by Vedantu is the gem book available on the internet. The solutions in it are sequenced in the format of Chapter wise and topic wise. So, students can easily search for the preparation of a particular topic of their interest and need. It provides solutions with explanations, short tricks, better illustrations, and special key points. While practising it, students can improve their problem-solving abilities.

5. What kind of books can be referred by Class 12 Mathematics students for getting good scores in the examination?

Many books are good for 12 Class students but NCERT books are very much helpful. After you are pretty confident with all problems in NCERT  you can try out sums from RD Sharma. This book just guides you to handle different kinds of problems. It also shows you different methods to solve a particular problem. It will take time to finish both the volumes of RDS but trust me after you finish it you will feel super confident and feel accomplished!