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You are given an indefinite integral to evaluate. What should you do?
Unfortunately, there is no algorithm one can use. Instead, there are perhaps hundreds of various tricks, most of them good for only a few specific integrals. Very few people today are as good integrators as masters of the past, in particular because technology freed us from that chore. Here we will do what is usual in these modern times: We restrict ourselves to the most important tricks that apply to relatively many integrals (we will therefore call them methods), they are based on methods covered in Theory - Methods of Integration and the reader would do well to read them before reading this. We will also introduce several useful types of integrals for which we do have reasonably reliable algorithms.
Back to the original question. We have an indefinite integral, what should we do? We have to choose the right procedure, by knowing the right signs and by experience. In order to learn integration one has to practice a lot and use it to build mental "boxes" of basic types of integrals. Then, when meeting a new integral. one tries to place it in the right "box". If the decision was right, then the method corresponding to the chosen box will solve the integral, or at least change it into an easier integral than might fall into another box for solving. We will now introduce the most important boxes.
Let's close this intro with a bit of optimism. Many integrals do not exactly fit into any box and then you have to try all kinds of dirty tricks. Some even cannot be solved at all. However, if the integral you face comes from school (for instance from a test), then it should be solvable (unless the examiner made a mistake in the question) and so it should fall to some box. If you do not give up, sooner or later you should break it.
Boxes for integration:
are the easiest since you remember them. Whenever you meet a new integral,
it is well worth to spend a few seconds by checking whether this integral is
one of those you know. This is all the more important because some of the
elementary integrals also perfectly fit into some box, which can be very
seductive. It sometimes happens that a student automatically applies an
appropriate procedure, after three pages of calculations obtains (with a
little bit of luck) the correct answer, and when seeing it realizes that it
could have been done on one line.
Many integrals can be also transformed into elementary integrals by some algebraic modifications, in general we often use linearity of integral. Note that algebraic simplifications are extremely useful even when it comes to integrals from other boxes, we will show some of the more useful now.
Substitution is the method of choice. It often changes ugly integrals into nice ones, sometimes even into elementary ones. Many substitutions can be done in the head without much writing, for instance a linear substitution (which always works but be careful about a sign then, another frequent mistake). When you see an integral, it pays to first inquire whether some nice substitution would not handle the problem. This applies even to integrals that exactly fit into some box below, because sometimes substitution offers an easier solution. For instance, there are rational functions that can be integrated on two pages using the standard procedure (see bellow), or on one line using substitution.
There are some signs that can point to substitution, some so prominent that an experienced integrator sees them at the first sight, we talk about it in the link above. But that is just scratching the surface, substitution is amazingly versatile and often helps in surprising (and devious) ways, here integration is close to art and a lot depends on inspiration. Do not worry, most likely you will not be required to perform such feats of high magic, what we do in Math Tutor should suffice.
Integration by parts is a borderline method. On one hand, it can surprise with its usefulnes, so it can be considered a more general method (just like substitution). On the other hand, this does not happen too often, in most cases it is applied to specific types of integrals, which makes it similar to boxes from the next paragraph.
For integrals of several types we have specialized boxes:
• integrals of rational functions, that is, ratios of polynomials, before going there it is a good idea to learn partial fractions decomposition,
• integrals with roots,
• integrals with ratios of linear expressions,
• trigonometric integrals, that is, integrals with sinces, cosines etc.; we also look at integrals with hyperbolic functions here,
• integrals with roots of quadratics.
If the integral does not fit any box, then you have got a problem. It is time for experiments. One possible approach to explore is some unusual substitution which perhaps would change the integral into something else and you start getting some ideas. If this does not work, then some tricky integration by parts might help. Or perhaps the integral can be rewritten in another form? After all, if you meet this integral at school, it should be solvable. However, when a student is well-prepared, these problems should not trouble him (most of the time).
Sometimes after applying the method from a suitable box you do not get an answer but a new integral to solve, preferably simpler. Then you apply the same procedure again and try to fit it into a suitable box, repeating this until the problem is solved (or you hit a dead end). The words "same procedure" in particular include checking on algebraic simplification, often the procedure you apply leaves the new integral in a form that is far from optimal.
Whichever way it goes, remember that the result of an indefinite integral
must include the integration constant (typically
Important note: Apart from linearity and methods described above, there are no other rules one could use, in other words, you have to curb your creativity. In particular, it is absolutely out of question to pull some expression with the variable out of the integral, there is also no product rule, so please do not split an integral of a product into a product of two integrals. True, you would not be the first to do so, but if you do, you will not be the last to get zero points for that.
Obviously, learning integration has two stages: First one has to learn well various methods and procedures from boxes, then one has to learn to fit integrals into the right boxes based on how they look. We offer a small selection of basic types here as an example.
In each of the boxes above we included typical examples. More examples of integration can be found in Solved Problems - Integrals, where you can moreover see other ways of evaluating integrals and various tricks. It is also the right place to see the main decision-making process in action. In fact, it might be a good idea to open this link in a separate window and look at it when learning methods from boxes, so that you can compare theory and real integrals. For practicing type recognition we offer the following idea: Go to Exercises, but do not really calculate those problems. For each integral, just make a guess which way should the solution go and then check the first hint whether you guessed right. If not, then try to think about what went wrong.
Remark on the set of validity of an integral. The requirements on this set are as follows: It should be a union of non-degenerate intervals, on each of them the expression in the answer must be continuous and on their interiors the derivative of the answer must give the function from the question.