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# RD Sharma Class 12 Chapter 19 Solutions (Indefinite Integrals)

RD Sharma Class 12 Maths Solutions for Chapter 19 Indefinite Integrals discusses Indefinite Integrals from its core. These RD Sharma Class 12 Solutions are very helpful for you to answer almost all questions that will be asked in CBSE and other competitive exams like JEE Main and JEE Advanced. The topics you will see in the chapter are Primitive or Antiderivative, Indefinite Integral, Fundamental Integration Formulas, some standard results on integration, Geometrical Interpretation of Indefinite integrals, Comparison between differentiation and integration, Methods of integration including substitution, parts, and rational algebraic functions by using partial fractions, integrals of the various forms, some special integrals, evaluation of the integrals, integrals reducible to the various forms, integrals to the various types, integrals by parts, some important integrals, and integration of rational algebraic functions by using partial fractions.

The chapter comprises 27 exercises with a total of 588 questions. These questions discuss every single topic that has been discussed in the chapter of Indefinite Integrals. All these questions have been answered by Instasolv in a very easy language for you to understand the usage of the concepts through logic and mathematical rules.

Our subject matter experts have prepared these Mathematics solutions with complete details for aspiring students like you. These solutions have helped countless students studying in CBSE 12th standard and those who are preparing to appear for competitive exams. These solutions are provided to you at no cost with 24×7 access.

## Some Topics discussed in RD Sharma Class 12 Maths Solutions Chapter 19 Indefinite Integrals

Definition of Indefinite Integrals

If f(x) is a function, then all its antiderivatives or primitives will be known as indefinite integrals of the function f(x), denoted by ∫f(x)dx.

Consider ∫f(x)dx as the indefinite integral of function f(x) w.r.t x.

The process by which we find the indefinite integral of a specific given function, it is said to be the integration of the function.

Fundamental Integration formulas

D/dx{Φ(x)} = f(x)

⇔ ∫f(x)dx = Φ(x) + C

Some of the integration formulas are

1. ∫ xn dx = ((xn+1)/(n+1))+C ; n≠1
2. ∫ sin x dx = – cos x + C
3. ∫ cos x dx = sin x + C
4. ∫ sec2 dx = tan x + C
5. ∫ csc2 dx = -cot x + C
6. ∫ sec x (tan x) dx = sec x + C
7. ∫ csc x ( cot x) dx = – csc x + C
8. ∫ (1/x) dx = ln |x| + C
9. ∫ ex dx = ex+ C
10. ∫ ax dx = (ax/ln a) + C ; a>0,  a≠1

Integration formulas are further classified into the following:

1. Irrational functions
2. Trigonometric functions
3. Rational functions
4. Logarithmic functions
5. Gaussian functions
6. Inverse trigonometric functions
7. Hyperbolic functions
8. Inverse hyperbolic functions
9. Exponential functions

Standard results on Integration

Theorem –

1. D/dx(∫f(x)dx) = f(x) i.e. an integral’s differentiation will be an integrand or integration and differentiation will be the inverse operations.
2. ∫k f(x) dx = k∫f(x) dx, where k will be a constant i.e. the integral of a (constant x function) will be the product of the constant and the function’s integral
3. ∫{f(x) ± g(x)}dx = ∫ f(x)dx ± ∫g(x)dx which means the integral of the sum of a finite number or a difference of a finite number of functions will be the sum of the integrals or the difference of the integrals of the various functions.

### Discussion of exercises in RD Sharma Class 12 Maths Solutions for ‘Indefinite Integrals’

• Exercise 19.1 tests your knowledge on evaluating integrations of a given function w.r.t x
• Exercise 19.2 will ask you to evaluate an integration of a given equation.
• Exercise 19.3 comprises 51 questions on integrating the given integrals.
• Exercise 19.4 has 51 questions where you’ll be asked to evaluate the given integrals.
• Exercise 19.5 has 72 questions where you will evaluate the given integrals of the form ∫{f(x)}nf’(x)dx
• Exercise 19.6 comprises eight questions on evaluating integrals of the form ∫ (ax+b)n P (x)dx, {P(x)/(ax+b)n}dx, given P(x) a polynomial and n a positive rational number
• Exercise 19.7 has 12 questions discussing the methodology to calculate trigonometric integrals.
• Exercise 19.8 has 13 questions talking about the methodology to calculate integrals.
• Exercise 19.9 has 10 questions where you will have to apply methodology to calculate some special integrals.
• Exercise 19.10 comprises 5 questions asking you to evaluate integrals of the type ∫(1/(ax2 + bx +c))dx
• Exercise 19.11 solves integrals that are reducible to ∫(1/(ax2+bx+c))dx the given 14 questions
• Exercise 19.12 consists of 9 questions solving the integrals of the type ∫(1/√ax2+bx+c)dx
• Exercise 19.13 comprises 17 questions of evaluating integrals that are reducible to the form ∫(1/√ax2+bx+c)dx
• Exercise 19.14 consists of 12 questions where you will evaluate integrals of the form ∫(px+q)/ (ax2+bx+c)dx
• Exercise 19.15 solves questions on evaluating the integrals of the form ∫(P(x)/ax2+bx+c)dx, where P(x) will be a polynomial of degree – 2 or higher.
• Exercise 19.16 talks about methodology in evaluating the integrals of the form ∫((px+q)/√ax2+bx+c)dx
• Exercise 19.17 asks questions based on integrals of the trigonometric form
• Exercise 19.18 discusses integrals of the form ∫(1/(a sin x +b cos x))dx, ∫(1 / (a + b sin x))dx, ∫(1/ (a sin x + b cos x + c))dx, ∫(1 / (a + b cos x))dx
• Exercise 19.19 talks about integrals of the form ∫((a sin x + b cos x + c) / (p sin x + q cos x + r))dx
• Exercise 19.20 discusses integration by parts and methodology to solve them in all its 58 questions.
• Exercise 19.21 discusses integrals of the form ∫eª {f(a) + f’(a)} da
• Exercise 19.22 talks about ways of solving ∫e^ax sin bx dx and ∫e^ax cos bx dx
• Exercise 19.23 comprises of questions that discuss methods and integrals like ∫√ax^2 + bx +c dx and ∫(px+q)√ax^2 + bx + c dx
• Exercise 19.24 consists of the questions to solve integrals of the form  ∫(px+q)√ax^2 + bx + c dx
• Exercise 19.25 will test your knowledge on integration of rational algebraic functions by using partial fractions
• Exercise 19.26 discusses questions on integrals of a particular form
• Exercise 19.27 discusses questions on integration of some special irrational algebraic functions.

### Benefits of Using Instasolv for RD Sharma Class 12 Maths Solutions Chapter 19 Indefinite Integrals

• Instasolv has employed subject matter experts with a rich experience in delivering easy to understand solutions on Indefinite Integrals.
• As the chapter talks about using Indefinite Integrals and its applications in other concepts of Class 12 Mathematics, Instasolv provides a different yet remarkable approach in solving these exercises.
• You can learn these chapters with minimum to no help if you follow the RD Sharma Solutions carefully. You will get all the solutions in exactly the same order as is found in Chapter 19 Indefinite Integrals. The best part is, these solutions are available for free.
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