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Edited By Kuldeep Maurya | Updated on Jan 24, 2022 02:22 PM IST

The RD Sharma books have gained popularity and are widely used by class 12 students for their reference purposes. However, chapter 19, the Definite Integrals portion, is always a challenging one for the students when it comes to mathematics. Therefore, to save the students’ time solving and rechecking the sums in the second exercise, Ex 19.2, the RD Sharma Class 12th Exercise 19.2 book will lend a helping hand.

Definite integrals exercise 19.2 question 2

Put

When x=1 then t=1 and when x=2 then t=1+log2

Definite integrals exercise 19.2 question 3

Putting

When x=1 then t=9-1=8 and when x=2 then t=36-1=35

Definite integrals exercise 19.2 question 4

Putting

Let

When x=0 then and when then

Definite integrals exercise 19.2 question 5

Putting

When x=0 then t=a and when x=a then t=2a

Definite integrals exercise 19.2 question 6

Put

When x=0 then t=1 and when x=1 then t=e

Definite integrals exercise 19.2 question 7

Put

When x=0 then t=0 and when x=1 then t=1

Definite integrals exercise 19.2 question 8

Put

When x=1 then t=log1=0 and when x=3 then t=log3

Definite Integrals exercise 19.2 question 9.

**Hint:** We use indefinite integral formula then put limits to solve this integral.**Given: ****Solution:**

Put

When x=0 then t=0 and

when x=1 then t=1

Definite Integrals exercise 19.2 question 10.

Put

When x=0 then and

when x=a then

Definite Integrals exercise 19.2 question 11.

Put

When then and when then

Definite Integrals exercise 19.2 question 12.

Put

When x=0 then t=0 and when then t=1

Definite Integrals exercise 19.2 question 13.

Put

When then and

When then

Definite Integrals exercise 19.2 question 14.

Put

When x=0 then t=3 and

when then

Definite Integrals exercise 19.2 question 15.

Put

When x=0 then t=0 and

when x=1 then

Definite Integrals exercise 19.2 question 16.

Put

When x=0 then t=2 and

when x=2 then t=2

Definite Integrals exercise 19.2 question 17.

Put

When x=0 then and

when x=1 then

Applying integration by parts method, then

Definite Integrals exercise 19.2 question 18.

Put

When x=0 then t=0 and when then t=1

Definite Integrals exercise 19.2 question 20.

Put

Put

When x=0 then t=0 and when then t=1

Definite Integrals exercise 19.2 question 21.

Equating coefficient of sinx and cosx respectively, we get

Solving equation (i) and (ii)

k-L=1

k+L=0

2k=1

k=12

ii=k+L=0

12+L=0

L=-12

Put

Definite Integrals exercise 19.2 question 22.

Put

Put

When x=0 then t=0 and when then

Definite Integrals exercise 19.2 question 23.

Applying integration by parts method we get

Put

Then

Definite Integrals exercise 19.2 question 24.

Applying integration by parts method we get

In put

Definite Integrals exercise 19.2 question 25

Put

When x=0 then t=-1 and when then t=0

Definite Integrals exercise 19.2 question 26

Put

When x=0 then t=0 and when then t=1

Definite Integrals exercise 19.2 question 27

Put

Put

When x=0 then t=0 and then

Definite Integrals exercise 19.2 question 28

Dividing numerator and denominator by cos2x , we get

Put

When x=0 then t=0 and when then

Definite Integrals exercise 19.2 question 29

Applying integration by parts method to first part of integral then,

Definite Integrals exercise 19.2 question 30

Put

When x=0 then t=0 and when x=1 and

Definite Integrals exercise 19.2 question 31

Put

- if x =0 then t = 1 & if then t = 0

Definite Integrals exercise 19.2 question 32

Applying integration by parts, then,

Definite Integrals exercise 19.2 question 33

Dividing the num. and denom. by x

put then,

Definite Integrals exercise 19.2 question 34

put than,

when x=0 then t=1 & when x=1 then t=2 ,

Definite integrals exercise 19.2 question 35

Put

When x = 4 then t =0 & when x =12 then t = 2

Definite integrals exercise 19.2 question 36

Applying integration by parts method, then,

On applying integration by parts method

Definite integrals exercise 19.2 question 37

put

when

therefore ,

Definite integrals exercise 19.2 question 38

put x+1x=t⇒1-1x2dx=dt

when x=0 then t= ∞ ,when x=1 then t=2

Therefore ,

Definite integrals exercise 19.2 question 39

put

when x= -1 then t=0 and when x=1 then t=2

therefore,

Definite integrals exercise 19.2 question 40

on multiplying and dividing by sec

put

when x=0 then t=0 , when then

therefore ,

To solve this integral, first we need to find its partial fraction then integrate it using indefinite integral formula then put the limits to get required answer.

therefore ,

Equating the coefficient of t3,t2,t and constant term respectively then

… a

… b

… c

…(d)

-4A=C

-4B=D

Since A= -C=0

Substracting (d) by (b), we get

4B+D=0

B+D =13B= -1

B=-13

1= -13+D ? 1+13=D?3+13=D

Therefore

Therefore

Now (i)?

put 2t=u ⇒2dt=du ⇒ dt=du2 in second integral, then

Definite integrals exercise 19.2 question 41

Definite Integrals exercise 19.2 question 42.

Put

when then t=1 and when then t=9

Therefore

Definite Integrals exercise 19.2 question 43.

put

when then and when then

Therefore ,

Definite Integrals exercise 19.2 question 44.

put

when x=0 then t=0 when then

Therefore

Definite Integrals exercise 19.2 question 45.

put

when x=1 then t=1 when x=2 then t=1+log2

Definite Integrals exercise 19.2 question 46.

Put

when x=0 then t=0

when then t=1

Definite Integrals exercise 19.2 question 47

Put

when x=4 then t=22

when x=9 then t=3 then

Definite Integrals exercise 19.2 question 48.

put

When x=0 then t=1

when then t=-1

Definite Integrals exercise 19.2 question 49.

Put

when x=0 then t=0 and

when , then t=1

Applying integration by parts, method then

Definite Integrals exercise 19.2 question 50.

put

when x=0 then t=0 &

when then t=1

Applying by parts, then

Applying integration by parts, then

When x=0 then

and when x=1 then t=0

therefore,

Definite Integrals exercise 19.2 question 52.

Putting

When and

Again putting

When then u=0

and then u=1

Applying integration by parts, method then

Definite Integrals exercise 19.2 question 53.

Definite Integrals exercise 19.2 question 54.

Put

When x=0 then & when x=a then

Definite Integrals exercise 19.2 question 55.

Put

When x=-a then & when x=a then

Definite Integrals exercise 19.2 question 56

To solve this integral, first we need to find it’s partial fraction then integrate it by using indefinite formula.

Equating coefficient of t and constant resp. , then

1=A+B ...a

0=2A+B⇒B=-2A ...b

Put the value of B in a

1=A+-2A=A-2A=-A

A=-1

From a⇒A+B=1⇒-1+B=1⇒B=1+1=2

∴A=-1,B=2

put

when x=0 then t=0 and then t=1

Therefore ,

Definite Integrals exercise 19.2 question 58

put

where x=0 then u=0 and when then

Therefore,

again , put

when u=0 then t=0 and when then

Therefore ,

Definite Integrals exercise 19.2 question 59

Put

When then t=8 and when x=1 then t=0

Definite Integrals exercise 19.2 question 60

Dividing numerator and denominator by

Put

Now x=0 then t=0 & when then t=1

Again putting

then

Definite Integrals exercise 19.2 question 61

Put

Now x=0 then t=1 & when then t=0

Definite Integrals exercise 19.2 question 62

Put

when

Class 12, mathematics, chapter 19, Definite Integrals, is a challenging portion where the students get frequent confusion and lose marks. Exercise 19.2 consists of the concept of evaluating the Definite Integrals. There are 62 questions to be solved in this exercise. This set of questions is divided into 49 questions in Level 1 and the remaining 13 questions in Level 2. The students can refer to the RD Sharma Class 12 Chapter 19 Exercise 19.2 material to gain ideas and solve the sums without confusion.

Experts from the educational sector and people with in-depth knowledge in the respective domain have contributed to preparing the solutions in RD Sharma books. It follows the NCERT pattern; this is why it is recommended by most of the CBSE board schools to their students. The solutions in the RD Sharma Class 12th Exercise 19.2 consist of problems solve in every possible method. This gives the freedom for the students to select the way that they feel is easy to adapt.

Definite Integrals is not a chapter that can never be solved; all required is proper practice with a good set of reference materials. For example, the Class 12 RD Sharma Chapter 19 Exercise 19.2 Solution book, consists of various practice questions, excluding the solutions given in the textbook. Once a student works on these additional questions, they tend to get familiarised with the concept.

It is a boon that the RD Sharma Class 12 Solutions Definite Integrals Ex 19.2 are available for students for free of cost at the Career 360 website. They need not spend money to purchase other solution books. The RD Sharma books will serve all the needs. Moreover, the RD Sharma Class 12th Exercise 19.2 books can also be downloaded from the same Career 360 website without monetary payment.

The previous batch students have benefitted a lot by using the RD Sharma Solutions Chapter 19 ex 19.2 to prepare this portion for their exams. As there are high chances of sums being asked from the RD Sharma book, it is wise to prepare with these books from the first day of exam preparation.

- Chapter 1 - Relations
- Chapter 2 - Functions
- Chapter 3 - Inverse Trigonometric Functions
- Chapter 4 - Algebra of Matrices
- Chapter 5 - Determinants
- Chapter 6 - Adjoint and Inverse of a Matrix
- Chapter 7 - Solution of Simultaneous Linear Equations
- Chapter 8 - Continuity
- Chapter 9 - Differentiability
- Chapter 10 - Differentiation
- Chapter 11 - Higher Order Derivatives
- Chapter 12 - Derivative as a Rate Measurer
- Chapter 13 - Differentials, Errors and Approximations
- Chapter 14 - Mean Value Theorems
- Chapter 15 - Tangents and Normals
- Chapter 16 - Increasing and Decreasing Functions
- Chapter 17 - Maxima and Minima
- Chapter 18 - Indefinite Integrals
- Chapter 19 - Definite Integrals
- Chapter 20 - Areas of Bounded Regions
- Chapter 21 - Differential Equations
- Chapter 22 - Algebra of Vectors
- Chapter 23 - Scalar Or Dot Product
- Chapter 24 - Vector or Cross Product
- Chapter 25 - Scalar Triple Product
- Chapter 26 - Direction Cosines and Direction Ratios
- Chapter 27 - Straight Line in Space
- Chapter 28 - The Plane
- Chapter 29 - Linear programming
- Chapter 30- Probability
- Chapter 31 - Mean and Variance of a Random Variable

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Download E-book1. Where can I find the RD Sharma books for Class 12 Mathematics, chapter 19?

The RD Sharma Class 12th Exercise 19.2 solution is available at the Career 360 website for free of cost. Anyone can access this set of solution books for various classes and subjects.

2. What are the advantages for the students who use the RD Sharma solution books for reference?

The solutions given in this book are prepared by experts.

The solutions are given in various methods.

Numerous practice questions are given.

3. Are the solutions provided in the RD Sharma books verified?

The solutions given in the RD Sharma books are provided by the experts and verified for accuracy. Therefore, the students need not have any hesitation regarding it.

4. Do the RD Sharma books contain solutions for Level and Level 2 questions for chapter 19, mathematics?

The RD Sharma Class 12th Exercise 19.2 solution book consists of answers for the Level 1 and Level 2 questions given in the textbook.

5. Can I download the RD Sharma books?

Yes, the option to download the RD Sharma solution books is given on the Career 360 website.

Mar 22, 2023

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