# RS Aggarwal Class 12 Chapter 13 Solutions (Methods of Integration)

RS Aggarwal Solutions for Class 12 Chapter 13 ‘Methods of Integration’ are of crucial importance while preparing for the school exams and other competitive exams. The answers to these questions will cover all the techniques of integration. The techniques in your syllabus are Integration by Substitution, Integration by Parts, Integration using trigonometric identities, Integration by some particular functions, Integration using partial fractions.

You will get the answers of 86 questions in 3 exercises in this chapter. There is in-depth coverage of all the topics in RS Aggarwal Solutions for Class 12 Chapter 13 ‘Methods of Integration’. These questions will help you stay a step ahead among your batchmates. Not only practising these solutions will help you achieve good scores and ranks it will also prepare a robust base in Calculus for you. The questions range from easy to tough of which the answers are mentioned each, adhering the guidelines given by the Board.

The subject matter experts at Instasolv have created these answers in an arranged manner to enable you to adapt to the pace of increasing complexity in the questions. The maths faculty of Instasolv is well versed and experienced in the most recurring doubts of students in this topic. Hence you can rest assured that all your queries will be taken care of at Instasolv in an inclusive and interactive way.

## Topics Covered in RS Aggarwal Solutions for Class 12 Chapter 13 Methods of Integration

**Introduction:**

Integration is the process of adding values on a large scale, where we can’t carry out general addition. However, there are many techniques of integration, used to integrate functions. There are various integration techniques that can be used to obtain an integral of some function, which is easier to evaluate the original integral. Here we will discuss the various techniques of integration such as integration by parts, integration by substitution, integration by partial fractions in detail.

**Integration Methods**

The different methods of integration include:

- Integration by Substitution
- Integration by Parts
- Integration Using Trigonometric Identities
- Integration of Some particular function
- Integration by partial fraction

**Integration By Substitution**

It can often be very difficult to find the integration of a function. In this method, we can obtain the integration by introducing a new independent variable. This method is known as Integration By Substitution.

The form of integral function at hand (say ∫f(x)) can be converted into another by changing the independent variable from x to t,

putting x = g(t) in the function ∫f(x), we have;

implies that, dx = g'(t).dt

Thus, I = ∫f(x).dx = f(g(t)).g'(t).dt

**Integration By Parts**

Integration by parts requires a special technique for integration of a function, where the integrand function is the multiple of two or more functions.

If f(x).g(x) be an integrand function then

Mathematically, integration by parts can be represented as.

∫ f(x) g(x) dx = f(x) ∫ g(x) dx – ∫ (f ′(x) ∫ g(x) dx) dx

For choosing the first and the second functions, we can follow the ILATE rule for integration.

**Integration Using Trigonometric Identities**

If the integrand uses any kind of trigonometric function, in the integration of a function, then we use trigonometric identities to simplify the function that can be easily integrated.

Few of the trigonometric identities are as follows:

**Integration of Some particular function**

The integration of a few certain types of functions involves some key formulae of integration that can be used to convert other integration into the standard form of the integrand. The integration of these standard integrands can be found using a direct form of integration method easily.

**Integration by partial fraction**

We know that a Rational Number can be expressed in the form of p/q, where p and q are integers and q≠0. In a similar way, a rational function is defined as the ratio of two polynomials which can be expressed in the form of partial fractions: P(x)/Q(x), where Q(x)≠0.

**Highlights:**

Forms of partial fractions:

- Proper partial fraction: It is of this form if the degree of the numerator is less than the degree of the denominator.
- Improper partial fraction: It is of this form if the degree of the numerator is greater than the degree of denominator then the fraction is known as an improper fraction.

### Exercises Discussion of RS Aggarwal Solutions for Class 12 Chapter 13 – Methods of Integration

- In the first exercise 13A, you will be required to evaluate the integration of various functions such as exponential functions, trigonometric functions etc using the method of substitution.
- In the exercise solutions of 13B, you will get questions where you will have to evaluate the integration of a diverse set of trigonometric functions
- In the exercise 13C, you will get miscellaneous questions to integrate the functions using trigonometric identities besides integrating some logarithmic functions.
- You will learn short tricks or quick approaches to such problems once you are done solving these questions.
- Practising these questions will also enhance your time management skills for the national level competitive entrance exams.

## Benefits of RS Aggarwal Solutions for Class 12 Maths Chapter 28 by Instasolv

You will be able to grasp the gist of the concepts of the Integration by studying these simplest possible solutions by the maths faculty of Instasolv. The subject experts at Insatsolv have written all the answers complying to the board guidelines. It is highly recommended that you supplement your coursebook with these RS Aggarwal Class 12 Maths Solutions by Instasolv. We provide you with a free of cost, simple to use student dashboard to help you make the most of your study hours.