One of the major sections of JEE preparation is Calculus. This includes Limits & Continuity, Differentiation calculus, Integration, Differential Equations & their real life applications (Area under curve etc.). Of all these, Integration is my personal favourite. Luckily enough, JEE likes it even more. They have been very versatile in asking problems from integral calculus. And, they always ask more than just one. There is a question from both definite and indefinite integration problems for sure.
Now, all said above (believe me on this!) is a great news. More questions they ask on integration, more you have a chance to score. There is a catch however; you have to practise really hard. Let me share powerful tools & advices given to me during my JEE prep.
Choose the right book: Carefully select your study material. Shanti Narayan, according to me is the best book for integral calculus. It constitutes a large variety of problems. You may also follow GN Berman. It should go without saying that you practise problems from NCERT books. Questions were asked in JEE from NCERT without much change! Try and solve mixed problem set once you are confident about your preparation.
Multiply & Divide, Add & Subtract: There is a fairly popular method known as Method of Partial Fractions (MoPF). However, it can be rather time consuming at times. Let me illustrate with a simple problem. Consider 
If you approach the problem using the MoPF it can be tedious. However, just add and subtract 1/2 in the numerator or you may also divide and multiply by 2. You’ll know what happens.
Break the numerator: Consider
Again, MoPF will be a inefficient method to solve it. Just separate out x & 1 in numerator to solve this simply. Illustration above is a rather simple example of this tool. I consider this as the most powerful method of solving such integrals. I can bet you will be coherent once you’ve solved a problem using this.
If the problem looks complicated, 99% of the times it is a blessing in disguise: It’s true if you can solve the problem you’ll score. But, you should know the motivation behind the problem. There is no need to panic if an ugly looking problem presents itself, keep calm. Trust the examiner and look for a derivative of a function in the problem itself. There are two very standard forms I am sure you have come across. Consider the forms
Think what’s their solution. Hint: It does not depend on f(x) & think ‘Integration By parts’.
Derive: Do you remember teachers asking you to derive textbook theorems for a better understanding. You may or may not have derived those while you were junior. However, to get grasp of integral calculus, it is best you derive simple integral formulae for ∫cot(x)dx, ∫ln(x)dx, ∫sec(x)d(x) etc. This is not just a mechanical exercise, it gives you a feel of how to approach a problem since you know the roots.
To Ponder
Integration began with the idea of obtaining area under a curve. The area curve was perceived as being sum of areas enclosed by infinitesimally small rectangles (vertical) of width 1/n & height as the value of curve at that ‘x’ (See figure-1). There is small difference between the area under curve and those of rectangles. This difference however becomes very small when n tends to ∞. So the sum of areas of rectangles is area under the curve.
So, Evaluate 
where x changes in steps of 1/n & n tends to ∞.
This is not a problem of Series & Sequences or of Limits & Continuity. It is highly likely to be a problem of Integral calculus & to evaluate, you just have to obtain ∫f(x)dx between x = a & x= b, f should be continuous.
All in all, Integration is a high scoring section. If you practise enough, you shall prevail!
