# RD Sharma Solutions for Class 12 Maths Chapter 19 Indefinite Integrals

## RD Sharma Solutions for Class 12 Maths Chapter 19 – Free PDF Download

RD Sharma Solutions for Class 12 Maths Chapter 19 –Â Indefinite IntegralsÂ is given here. By solving exercise-wiseÂ problems usingÂ RD Sharma Solutions for Class 12Â daily helps students improve their problem solving and logical thinking skills, which are important to achieve a better academic score. The main aim is to help students self analyze the areas, which require more practice from the exam perspective. With the help of RD Sharma Solutions, students can now solve the exercise problems in a shorter duration with a clear idea about the concepts.

The 19th Chapter, Indefinite Integrals of RD Sharma Solutions for Class 12 Maths explains some standard results on integration along with fundamental integration formulae.Â  TheÂ RD Sharma Solutions for Class 12 are formulated by BYJUâ€™S experts to provide a fundamental aspect of Maths, which in turn, assists students to understand every concept clearly.Â The solutions PDF is a major reference guide to help students score well in the Class 12 examination.

## Download the PDF of RD Sharma Solutions For Class 12 Maths Chapter 19 Indefinite Integrals

### Exercise 19.1 Page No: 19.4

1. Evaluate the following integrals:

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Given

2. Evaluate:

Solution:

Given

Solution:

Given

3. Evaluate:

Solution:

Given

### Exercise 19.2 Page No: 19.14

Evaluate the following integrals (1 – 44):

Solution:

Solution:

Given

Solution:

Given

Solution:

Given,

âˆ«(2 â€“ 3x)(3 + 2x)(1 â€“ 2x) dx

= âˆ«(6 + 4x â€“ 9x â€“ 6x2)(1 â€“ 2x) dx

= âˆ«(6 â€“ 5x â€“ 6x2)(1 â€“ 2x) dx

= âˆ«(6 â€“ 5x â€“ 6x2 â€“ 12x + 10x2 + 12x3) dx

= âˆ«(6 â€“ 17x + 4x2 + 12x3) dx

Upon splitting the above, we have

= âˆ«6 dx â€“ âˆ«17x dx + âˆ«4x2 dx + âˆ«12x3 dx

On integrating using formula,

âˆ«xn dx = xn+1/n+1

we get

= 6x â€“ 17/(1+1) x1+1 + 4/(2+1) x2+1 + 12/(3+1) x3+1 + c

= 6x â€“ 17x2/2 + 4x3/3 + 3x4 + c

Solution:

Given

Solution:

Solution:

Given

Solution:

Given

Solution:

Solution:

Given

Solution:

Given

Solution:

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Given

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Given

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Given

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Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

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Given

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### Exercise 19.7 Page No: 19.38

Integrate the following integrals:

Solution:

Solution:

Solution:

### Exercise 19.8 Page No: 19.47

Evaluate the following integrals:

Solution:

Solution:

Given,

Solution:

Given,

Solution:

Solution:

Solution:

Therefore,

= cos (b – a)x + sin(b – a) log |sin(x – b)| + c, where c is an arbitrary constant.

### Exercise 19.9 Page No: 19.57

Evaluate the following integrals:

dx

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

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### Exercise 19.11 Page No: 19.69

Evaluate the following integrals:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

### Exercise 19.14 Page No: 19.83

Evaluate the following integrals:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

By using,

Solution:

Solution:

### Exercise 19.16 Page No: 19.90

Evaluate the following integrals:

Solution:

Solution:

Solution:

Solution:

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### Exercise 19.17 Page No: 19.93

Evaluate the following integrals:

Solution:

Solution:

Solution:

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### Exercise 19.18 Page No: 19.98

Evaluate the following integrals:

Solution:

Solution:

Solution:

Solution:

Let sin x = t

Solution:

Solution:

Solution:

Solution:

### Exercise 19.19 Page No: 19.104

Evaluate the following integrals:

Solution:

We will solve I1Â and I2Â individually.

Solution:

Solution:

Solution:

Solution:

### Exercise 19.20 Page No: 19.106

Evaluate the following integrals:

Solution:

Solution:

â‡’Â 1 = (A + B) x + (3A â€“ 2B)

â‡’Â Then A + B = 0 â€¦ (1)

And 3A â€“ 2B = 1 â€¦ (2)

Solving (1) and (2),

2 Ã— (1)Â â†’Â 2A + 2B = 0

1 Ã— (2)Â â†’Â 3A â€“ 2B = 1

5A = 1

âˆ´Â A = 1/5

Substituting A value in (1),

Or I = log|(x – 2)/(x + 3)| + x + c

Solution:

Solution:

Solution:

Hence,

### Exercise 19.21 Page No: 19.110

Evaluate the following integrals:

Solution:

Solution:

Solution:

Solution:

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### Exercise 19.22 Page No: 19.114

Evaluate the following integrals:

Solution:

Solution:

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### Exercise 19.23 Page No: 19.117

Evaluate the following integrals:

Solution:

Solution:

Solution:

Solution:

5.

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### Exercise 19.24 Page No: 19.122

Evaluate the following integrals:

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### Exercise 19.25 Page No: 19.133

Evaluate the following integrals:

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### Exercise 19.26 Page No: 19.143

Evaluate the following integrals:

Solution:

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Solution:

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### Exercise 19.27 Page No: 19.149

Evaluate the following integrals:

Solution:

Solution:

Solution:

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### Exercise 19.28 Page No: 19.154

Evaluate the following integrals:

Solution:

Solution:

Solution:

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### Exercise 19.29 Page No: 19.158

Evaluate the following integrals:

Solution:

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### Exercise 19.30 Page No: 19.176

Evaluate the following integrals:

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### Exercise 19.31 Page No: 19.190

Evaluate the following integrals:

Solution:

The given equation can be written as,

Solution:

Now, substituting t as x â€“ 1/x and z as x + 1/x we have

Solution:

Solution:

We get,

Solution:

### Exercise 19.32 Page No: 19.196

Evaluate the following integrals:

Solution:

Solution:

Solution:

Solution:

Solution:

### Access all the exercises of RD Sharma Solutions For Class 12 Chapter 19 â€“ Indefinite Integrals

Exercise 19.1 Solutions

Exercise 19.2 Solutions

Exercise 19.3 Solutions

Exercise 19.4 Solutions

Exercise 19.5 Solutions

Exercise 19.6 Solutions

Exercise 19.7 Solutions

Exercise 19.8 Solutions

Exercise 19.9 Solutions

Exercise 19.10 Solutions

Exercise 19.11 Solutions

Exercise 19.12 Solutions

Exercise 19.13 Solutions

Exercise 19.14 Solutions

Exercise 19.15 Solutions

Exercise 19.16 Solutions

Exercise 19.17 Solutions

Exercise 19.18 Solutions

Exercise 19.19 Solutions

Exercise 19.20 Solutions

Exercise 19.21 Solutions

Exercise 19.22 Solutions

Exercise 19.23 Solutions

Exercise 19.24 Solutions

Exercise 19.25 Solutions

Exercise 19.26 Solutions

Exercise 19.27 Solutions

Exercise 19.28 Solutions

Exercise 19.29 Solutions

Exercise 19.30 Solutions

Exercise 19.31 Solutions

Exercise 19.32 Solutions

## RD Sharma Class 12 Maths Solutions Chapter 19Â Indefinite Integrals

Some of the essential topics covered in this chapter are listed below.

• Definition of primitive or antiderivative
• Definition and meaning of indefinite integral
• Fundamental integration formulae
• Some standard results on integration along with the corollary
• Integration of trigonometric functions
• Integration of exponential functions
• Miscellaneous problems
• Geometrical interpretation of indefinite integral
• Comparison between differentiation and integration
• Methods of integration
• Integration by substitution
• Some standard results
• Evaluation of integrals by using trigonometric substitutions
• Some special integrals
• Integration by parts
• Some important integrals along with theorems
• Integration of rational algebraic functions by using partial fractions
• When the denominator is expressible as a product of distinct linear factors
• When the denominator contains some repeating linear factors
• The denominator contains irreducible quadratic factors
• Integration of some special irrational algebraic functions

## Frequently Asked Questions on RD Sharma Solutions for Class 12 Maths Chapter 19

### How BYJUâ€™S RD Sharma Solutions for Class 12 Maths Chapter 19 help the students in preparing for board exams?

RD Sharma Solutions for Class 12 Maths Chapter 19 helps the students in gaining a better knowledge of the topics covered. Our solution module uses various examples and diagrams to explain the questions, wherever necessary. For CBSE board students aiming at securing an excellent score, solving RD Sharma Solutions for Class 12 is a must. Solving the questions from each exercise will ensure that the students score high marks in the exams.

### Why should we follow RD Sharma Solutions for Class 12 Maths Chapter 19?

RD Sharma Solutions for Class 12 Maths Chapter 19 is the best reference material that offers complete and quality information about different Math concepts. The solutions have been solved in an easy-to-remember format, which further helps students to clearly understand and remember the answers. To score good marks, practising these solutions for Class 12 Maths can help to a great extent. Itâ€™s clear that RD Sharma Textbooks for Class 12 are essential reference books to score high in examinations.

### How can we score full marks using RD Sharma Solutions for Class 12 Maths Chapter 19?

The RD Sharma Solutions for Class 12 Maths Chapter 19 are formulated by experts at BYJUâ€™S after conducting extensive research on each concept. Every minute detail is explained in a comprehensive manner to help students score well in the class test as well as board exams. It also helps students in doing their assignments given to them on time without any difficulty.