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RD Sharma Class 12 Solutions Chapter 19 - Indefinite Integrals (Ex 19.5) Exercise 19.5

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RD Sharma Class 12 Solutions Chapter 19 - Indefinite Integrals (Ex 19.5) Exercise 19.5 - Free PDF

Download of RD Sharma Class 12 Solutions Chapter 19 - Indefinite Integrals Exercise 19.5 solved by Expert Mathematics Teachers on Vedantu.com. All Chapter 19 - Indefinite Integrals Ex 19.5 Questions with Solutions for RD Sharma Class 12 Maths 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.

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RD Sharma Class 12 Solutions Chapter 19 - Indefinite Integrals

More About The Chapter

‘Indefinite Integrals’ is Chapter 13 of RD Sharma Class 12 Math which is also known as the anti-differentiation as it is the opposite of differentiation. There are two types of integral- definite and indefinite. The indefinite integral of a function F is a differentiable function F, the derivative of which is equal to f, which is the original function. 


F'=f


Indefinite Integrals generally don’t have any upper or lower limits.


\[\int f(z)dz\]


The indefinite Integral formula is:


\[I= \int x^{n}. dx = \frac{x^{n+1}}{n+1} +c\]


Formulas Used in Indefinite Integrals


  1. \[\int x^{n}. dx = \frac{x^{n+1}}{n+1} +c\]

  2. \[\int 1. dx = x+c\]

  3. \[\int e^{x}. dx = e^{x}+c\]

  4. \[\int \frac{1}{x} dx = Log\left | x \right |+c\]

  5. \[\int a^{x}dx = \frac{a^{x}}{log a} + c\]

  6. \[\int \cos x.dx = \sin x +c\]

  7. \[\int \sin  x.dx = -\cos x +c\]

  8. \[\int \sec ^{2}x.dx = \tan x + c\]


Properties of Indefinite Integrals 


The two properties of Indefinite Integrals are-


1. The constant-coefficient can be taken outside the integral sign


\[\int a f(x).dx=a\int f(x)dx\]


Where a is constant.


2. If the functions are f(x) and g(x)


\[\int [f(x)\pm g(x)] = \int f(x)\pm \int g(x)\]


Difference Between Definite and Indefinite Integral

Definite Integral

Indefinite Integral

The definite integral of f(x) is a number.

The indefinite integral of f(x) is a function.

The upper limits and lower limits are constants.

There are no upper or lower limits.

A definite integral gives a definite answer at the end of the question.

Indefinite Integral will have a constant in it.

Definite Integrals with specific limits are possible to find.

Indefinite Integrals are not possible to find in closed forms.

Definite Integrals are widely used to calculate values of force, mass, work, surface areas, and so on.

Indefinite Integrals are used in Science where a general solution is required.

FAQs on RD Sharma Class 12 Solutions Chapter 19 - Indefinite Integrals (Ex 19.5) Exercise 19.5

1. Which is more widely used in real-life applications- Definite Integrals or Indefinite Integrals?

Both the Integrals have their own uses and importance in real-life practices. It is difficult to answer which one is more widely used. Indefinite Integrals are used to find displacement from velocity and velocity from acceleration using the definite integral while Definite Integrals can be used to calculate the force exerted on an object submerged in a liquid, to determine the mass of an object if its density function is known.

2. Why is there a need to add a constant c while doing indefinite integral calculations?

While solving a definite integral problem we need to make sure that all the solutions that can be possible are included. C is a constant value that we add to a solution which can be any number and can be 0 as well. We add a C because when we obtain a derivative all the constants are canceled out. C is the arbitrary constant and it allows us to express the general forms of derivatives. 

3. Why is RD Sharma Maths best for indefinite integral chapters in class 12?

Class 12 Maths helps the students to study efficiently and prepare them for higher studies. RD Sharma Class 12 Maths helps the students to understand the concepts clearly as the book contains conceptual problems along with solved examples. The indefinite integral is an important chapter since the differential equation is one of the most important topics to study in the subject of Mathematics. Students study this topic in other subjects as well such as physics and engineering, therefore it is very important to have clarity with the concepts. RD Sharma has designed the book in such a way that if a student goes through the topics thoroughly can have a good grasp on that topic.

4. How many exercises are there in RD Sharma Maths Class 12 Chapter 19 Indefinite Integrals?

There are a total of 32 exercises in RD Sharma Maths Class 12 Chapter 19. You can find the solutions to all the exercises on Vedantu. If the student wishes to study offline, he can download the pdf as well and choose to study anytime, anywhere. Vedantu solutions are easy to understand and help the students to understand the concept better as well.

5.How to study the Chapter Indefinite Integrals?

Before going for the Chapter Indefinite Integrals, one must have deep knowledge about integration. The student should then go through all the important formulas and concepts. For the students who find it difficult to learn from the book, Vedantu has provided revision notes in easy and understandable language. After learning all the concepts, students should then go for solving problems. Students should practice all types of questions from basic to high-level thinking questions in order to master the Chapter Indefinite Integrals.