So, here we are discussing one of the most crucial yet hard (or considered to be hard) topics of JEE mathematics. I hope after reading this blog, you will have enough insight on how to face integration problems asked in the JEE and also on how to deal with the integration problems in your regular JEE book. Don’t be afraid, let’s dive into this integration JEE world together.

## The Idea

Basically, integration means simply adding up pretty naively. However this adding up is done on infinitesimally (tends to 0) small intervals. It is also defined as the reverse process of differentiation. This kind of integration is done without any upper and lower limits, hence the name indefinite.

Where, C represents the integration constant which would not be present in case of definite integration.

## For the JEE

Now, basically the formulas for integration can be generally proved as the antiderivative.

So, by definition,

Similarly the others follow, and you can find them easily in any CBSE book–the best being NCERT mathematics parts one and two. Here are some of the special integrals which you should keep in your mind while solving the problems:

Now, let’s get to the content of JEE integration. The questions are usually based on the following three categories.

### The Powered Problems

To solve this type of problem, the first thing you should do is take the highest power of x (in this case 4) outside. The problem is designed in such a way that the differential of the term inside the bracket will be 1 + (1/x^{4}), which is exactly equal to the term outside the bracket 1/x^{5} (with maybe some considerations of constant terms).

### The Substitution Problems

In this case, you should try to eliminate by substituting x with asin^{2/3} θ, so that a gets cancelled out from both numerator and denominator. Then it becomes a trivial problem. In case you see there is a term of x^{4} along with some other terms. Try to substitute x with tan θ. If there is only 1 + x^{4} in the denominator then consider dividing numerator and denominator by x^{2} terms and proceed by method 1.

### The Recursive Problems

Get a strong hold on this matter from now on, as this is the most frequently asked topic from indefinite integration in the JEE. What does this mean? Recursion, huh?

One probably hardly remembers what is and in the recursive approach you are asked to find what is .

Sounds interesting, let’s check out:

Or,

Therefore, we have successfully got our recurrence relation. If we know I_{0}, we can find I_{2} and then I_{4} and hence all the values that n can take.

## Best Books On JEE Indefinite Integration

So, after discussing some important problems on JEE integration, it’s time to learn about the materials required to strengthen your JEE integration skills. Here are the best books in the order of precedence that you should definitely follow:

**NCERT**: As we all know this is the starting point of our knowledge. Everyone who cracks the JEE agrees that NCERT has helped them in their preparation. However, the point is that you should not spend too much time on this book as you have to solve a lot of other books as well. Just skim through the solved exercises and memorize the formulae and that will be enough.**Cengage Calculus**: This book contains many solved examples on indefinite integration and will give you a good idea on problem-solving. The level of problems here range from very difficult to very easy. Do not waste your time on solving only the easy problems, they are practically useless for the JEE. They are only there to get you accustomed to the chapter, so try to move quickly to more difficult problems in the chapter.The book comes fully solved, so it won’t be a problem if you are preparing without coaching. Always remember to attempt the solved exercises before the unsolved ones. There’s a plethora of them there, so manage your time as you have to attempt other chapters as well.

**Integral Calculus by Amit M. Agarwal**: This is one of the most difficult books on indefinite integration and hence becomes the best book for the JEE. Some problems given in this book are far above the standards of JEE, however, they are only a few in number while the rest are brilliant problems for your JEE preparation.It starts by discussing all the methods of integration following examples that make understanding very easy. Recursion in this book is explained very well.

I may be repeating myself, but it is important for you to know that you should do recursion very well if you want to ace JEE integration. Along with these qualities, it comes with solved solutions for almost all of the problems. There are two sections, one MCQ (multiple choice questions) and the other section is for subjective questions. Few problems in the subjective part are out of the current JEE standards, and also they do not have solutions.

### Don’t Forget Coaching Materials

Apart from the general books, you can also refer to the coaching material given by different institutes. Though the theory given in these materials isn’t really great, the problems given there are generally quite good. They are sometimes really thought-provoking.

So if you use these materials, read the theory from the books mentioned above, and solve the exercises from these materials. One disadvantage is that they do not always come fully solved as they are meant to be discussed in the classes. However you should definitely attempt them if you feel you have enough time to solve the other chapters.

## Some Final Tips And Tricks

So, apart from what we discussed above, indefinite integration is all about memorizing the formulae the correct way and assembling them the right way. Keep those formulae fresh in your mind as well as the integration of

tan x = log(|sec x|), cosec x = log (|tan x/2|), cot x = log(|sin x|) and similarly some others.

### Integration by Parts

Another method which should have been mentioned much earlier is the Integration by parts. You should master it very soon. Don’t try to remember it by the formulae given in the book, you’ll find it extremely difficult. Instead, try to remember by the means of an example:

I know, it’s a bit difficult at first, but later on, you will be able to do it mentally. Always remember to integrate one term at first, then differentiate the other term, thus integrating the whole.

I hope you have got some ideas about indefinite integration and how to master it for the JEE. So, what are you waiting for? Go on, start your beautiful journey!

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