In this lesson we will begin to study the topic Undefined Integral, as well as examine in detail examples of solutions of the simplest (and not quite) integrals. In this article I will limit myself to the minimum of theory, and now our task is to learn how to solve integrals.
What do we need to know for successful mastering of the material? In order to cope with integral calculus you need to be able to find derivatives, at least, at the middle level. So, if the material is running, I recommend that you first carefully read the lessons How to find a derivative? and Derived from a complex function. It would be a good experience if you have a few dozen (better, a hundred) self-finded derivatives. At least, you should not be stumped by the task of differentiating between the simplest and most common functions. It would seem, what does derivatives have to do with it in general, if the article deals with integrals?! Well, here's the thing. The point is that the discovery of derivatives and the discovery of undefined integrals (differentiation and integration) are two reciprocal actions, such as addition/subtraction or multiplication/separation. Thus, without the skill (+ some experience) of finding derivatives, unfortunately, no further progress can be made.
In this regard, we need the following methodological materials: the Derivative Table and the Integral Table. The reference materials can be opened, downloaded or printed on the Mathematical Formulas and Tables page.
What is the difficulty in studying undefined integrals? If in derivatives there are strictly 5 rules of differentiation, a table of derivatives and a rather clear algorithm of actions, then in integrals everything is different. There are dozens of integration methods and techniques. And, if the method of integration was originally chosen incorrectly (i.e., you do not know how to solve it), then an integral can be "stabbed" literally for 24 hours, like a real rebus, trying to identify different techniques and tricks. Some people even like it. By the way, it's not a joke, I quite often had to hear from students an opinion like "I've never had any interest in deciding the limit or derivative, but the integrals - quite another thing, it's fascinating, there is always a desire to "hack" a complex integral. Wait. Enough black humor, let's move on to these most undefined integrals.
If there are so many ways to solve this problem soon, where to start studying undefined integrals? In my opinion, there are three pillars in integral calculus, or a kind of "axis" around which everything else rotates. First of all, we should have a good understanding of the simplest integrals (this article). Then, we need to study in detail the lesson Method of substitution in an undefined integral. IS THE MOST IMPORTANT TRICK! It may be even the most important article of all my articles on integrals. And, thirdly, it is necessary to get acquainted with the method of integration in parts, because it integrates a large class of functions. If you learn at least these three lessons, it will be "not two" anymore. You may "forgive" your ignorance of integrals from trigonometric functions, integrals from fractions, integrals from fractional-rational functions, integrals from irrational functions (roots), but if you "sit in a puddle" on the replacement method or the method of integration in parts, it will be very, very bad.