# RS Aggarwal Class 12 Chapter 15 Solutions (Integration Using Partial Fractions)

RS Aggarwal Solutions for Class 12 Chapter 15- Integration using Partial Fractions have been prepared to provide you assistance in solving and understanding the complex topics of this chapter. This topic of calculus has a wide range of applications in different subjects from daily life. You will learn how you can evaluate the integration of the product of two functions using the method of Integration by parts. You will also get aid in understanding the methodology applied in the Slate Rule for integration in the RS Aggarwal Solutions for Class 12 Chapter 15. You can use these answers to solve the questions in this chapter effortlessly.

There are 50 questions in 2 exercises 15A and 15B comprehensively covering the integration of almost all the possible kinds of products of two functions. Solving these questions with the help of the solutions provided by Instasolv will boost your proficiency in approaching a maths problem. This will not just make you more confident in Maths but also enhance your score significantly. The questions are arranged in the increasing order of difficulty level to help you polish your aptitude in the problems of Integration and progress parallelly with the advancement of the exercise problems.

The Maths faculty of Instasolv employ layman’s language to assist you in sorting your doubts with any further complications. You will find all the steps clearly explained by the Instasolv experts in these solutions for RS Aggarwal Class 12 Chapter 15. At Instasolv, you will learn to manage your time well during the exam hours. Also, Instasolv is strictly compliant to the pattern of questions asked in the exams which is why it is a highly reliable source of reference.

## Topics Covered in RS Aggarwal Solutions for Class 12 Chapter 15 Integration Using Partial Fractions

**Introduction:**

Integration By Parts formula is applied for integrating the product of two functions. This formula is applied to obtain the integrals by reducing them into standard forms. Like, to find the integration of x sin x we need to use this method. The integrand is the product of the two functions. The formula for this method is given by;

Apart from integration by parts, there are two methods that are used to perform the integration. They are:

- Integration by Substitution
- Integration using Partial Fractions

**Derivation of Integration By Parts Formula:**

If u(x) and v(x) are any two differentiable functions of a single variable y. Then, by the product rule of differentiation, we get;

We already know how to differentiate a product: if

The first term on the R.H.S. will be simplified as we are simply integrating what has been differentiated.

This formula is termed as integration by parts.

**ILATE Rule:**

In integration by parts, we have grasped that if the product of two functions is provided to us then we apply the required formula. The integral of the two functions is taken, by considering the left term as first function and second term as the second function. This method is called the Slate rule. Suppose, we have to integrate x ex, then we consider x as the first function and ex as the second function. So, the first function is selected in a way such that the derivative of the function can be easily integrated. Usually, we want the order of this rule to be based on some functions such as Inverse, Algebraic, Logarithm, Trigonometric, Exponent.

**Highlights:**

- Integration by parts can not be applied to functions of the form

∫ √x sin x dx.

- We do not add a constant when calculating the integral of the second function.
- Usually, if any function is a power of x or a polynomial in x, then we use it as the first function. Although, in cases where a function is an inverse trigonometric function or logarithmic function, then we take them as the first function.

### Exercise-Wise Discussion of Questions in RS Aggarwal Solutions for Class 12 Chapter 15 – Integration using Partial Fractions

- You will learn the basic applications of the formula for Integration of functions by parts in the exercise solutions of 15A.
- The exercise solutions of 15B contain more advanced questions where you will have to simplify a given function into a simpler form of products of two functions so that you can make use of the method of Integration using Partial Fractions.
- Solving these exercise questions will prepare you for important national level competitive exams such as JEE, BITSAT, NEET etc.
- You will be able to outshine your batchmates if you are consistent in practising the RS Aggarwal Class 12 Chapter 15 questions with respect to the clarity of concepts.
- Having solved these exercise questions rigorously, you will be a step closer towards entering the college to kickstart your dream career.

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