RD Sharma Class 12 Maths Solutions Chapter 19 - Indefinite Integrals

Mathematics is a subject that requires constant practice. After solving NCERT, students solve books like RD Sharma to improve their concepts. Sometimes they get stuck while solving the questions. To help students with all the questions, we have made the solutions to RD Sharma Solutions for Class 12 Mathematics Chapter 19 - Indefinite Integrals.

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All the solutions to RD Sharma 12 Mathematics have been provided on this website which are free to download in PDF format. The solutions are very easy to go through for students. If a student goes through the RD Sharma Solutions for Class 12 Mathematics Chapter 19 - Indefinite Integrals they will understand that even the most daunting of questions can be solved with enough determination and patience.

Class 12 RD Sharma Textbook Solutions Chapter 19 - Indefinite Integrals

Given below are the important topics covered in this Chapter:

• Concept of the antiderivative.

• Definition and meaning of indefinite integral.

• Integrand and Element of Integration.

• Geometrical Interpretation of Indefinite Integral.

• Comparison between differentiation and integration.

• Rules of Integration.

• Properties of Indefinite Integral.

• Derivation of basic integration formulae.

• Some standard and interesting results on indefinite integrals.

• Integration of trigonometric functions.

• Integration of exponential functions.

• Methods of integration.

• Evaluation of integrals by using trigonometric substitutions.

• Various trigonometric identities are used for the conversion of Integrals into Integrable Forms.

• Method of Substitution.

• Standard Substitutions in integration problems.

• Integration by parts.

• Some special integrals.

• Some important integrals along with theorems.

• Partial fractions for the integration of rational algebraic functions.

• The denominator can be expressed as a product of distinct linear factors.

• The denominator has some repeating linear factors.

• The denominator contains non-reducible quadratic factors.

• Integration of some special irrational algebraic functions.

RD Sharma Class 12 Indefinite Integrals

We have provided step-by-step solutions for all exercise questions given in the PDF of Class 12 RD Sharma Chapter 19 - Indefinite Integrals. All the Exercise questions with solutions in Chapter 19 - Indefinite Integrals are given below:

Exercise 19.1

Exercise 19.2

Exercise 19.3

Exercise 19.4

Exercise 19.5

Exercise 19.6

Exercise 19.7

Exercise 19.8

Exercise 19.9

Exercise 19.10

Exercise 19.11

Exercise 19.12

Exercise 19.13

Exercise 19.14

Exercise 19.15

Exercise 19.16

Exercise 19.17

Exercise 19.18

Exercise 19.19

Exercise 19.20

Exercise 19.21

Exercise 19.22

Exercise 19.23

Exercise 19.24

Exercise 19.25

Exercise 19.26

Exercise 19.27

Exercise 19.28

Exercise 19.29

Exercise 19.30

Exercise 19.31

Exercise 19.32

In 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 have to find a function whose differential is given. So we can say, integration is a process that is the inverse of differentiation.

Note: A function can have infinite anti-derivatives of integrals whereas the derivative of a function is unique.

Properties of Indefinite Integral

(i) ∫

f(x)+g(x)

f(x)+g(x) dx = ∫f(x) dx + ∫g(x) dx

(ii) For any real number k, ∫k f(x) dx = k∫f(x)dx. (We can simply take out the constant from integration)

Using the above two properties, in general if f1, f2,………, fn are functions and k1, k2,…, kn are real numbers, then

∫ [k1f1(x) + k2f2(x) + … +  knfn(x)] dx = k1 ∫f1(x) dx + k2 ∫ f2(x) dx + … + kn ∫fn(x) dx.

Integration by Substitutions

We should use the Substitution method when a suitable substitution of variables leads to simplification of the integral.

If I = ∫f(x)dx, then put  x = g(z), we get

I = ∫f

g(z)

g(z) g'(z) dz

Note: In this method try to substitute the variable whose derivative is present in the original integral.

Integration by Parts

Let f(x) and q(x) be two functions, then we have

∫

f(x)q(x)

f(x)q(x) dx = f(x)∫g(x)dx – ∫{f'(x) ∫g(x)dx} dx

Here, we should choose the first function according to its position in ILATE to simplify our calculation, where

I = Inverse trigonometric function

L = Logarithmic function

A = Algebraic function

T = Trigonometric function

E = Exponential function

The function which comes first in ILATE should be taken as the first function and the other as the second function.

Integration by Partial Fractions

A rational function is the ratio of two polynomials of the form p(x)/q(x), where p(x) and q(x) are polynomials in x and   q(x) ≠ 0. If degree of p(x) > degree of q(x), then we can write p(x)/q(x)= t(x) + p1(x)/q(x), where the degree of p1(x) is less than the degree of q(x).

Since t(x) is a polynomial in x, therefore it can be integrated easily.

Now p1(x)/q(x) can be integrated by expressing p1(x)/q(x) as the sum of partial fractions of the following types:

i)p(x)+q(x−a)(x−b)=A(x−a)+B(x−b)

ii)p(x)+q(x−a)2=A(x−a)+B(x−a)2

iii)p(x)2+q(x)+r(x−a)(x−b)(x−c)=A(x−a)+B(x−b)+C(x−c)

iv)p(x)2+q(x)+r(x−a)2(x−c)=A(x−a)+B(x−a)2+C(x−b)

v)p(x)2+q(x)+r(x−a)(x2+bx+c)=A(x−a)+Bx+C(x2+bx+c)

where x2+ bx+c cannot be factored further.

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Conclusion:

A large number of practice problems are provided by Class 12 Mathematics RD Sharma Solutions Chapter 19 which helps the students in grasping the concepts and understanding the topic in a better way. At the end of every Chapter, Miscellaneous Exercise is given by RD Sharma’s book. The Miscellaneous Exercise problems have questions involving multiple topics that are covered in the Chapter. To solve Miscellaneous Exercise problems students must understand all the concepts related to this Chapter clearly. Solving these problems will give an extra advantage to students in exams.