Section 11: Methods of Integration
© 2000 by Karl Hahn
In the last section you saw a table, which you could use to look up the indefinite integrals of five different functions. Of course there is much more to calculus than just five functions. So if you are presented with a function that doesn't happen to be on that table, how are you to find its antiderivative?
When you learned to take derivatives, you learned a small set of rules that worked every time. If a function was a product, you knew to apply the product rule. If a function was a composite, you knew to apply the chain rule. If a function was a quotient, you knew to apply the quotient rule. The bad news with integration is that there is no product rule, no chain rule, and no quotient rule.
You have probably played with those solid wooden Japanese puzzles with interlocking pieces. When you start, it's the shape of a cube or a ball or perhaps an animal. Then you search around for the key piece that you can remove. Once you've pulled that one out, you can remove the remaining pieces one by one. Indeed once you have that first piece removed, it's abundantly clear how to take the puzzle apart.
Taking the puzzle apart -- that's what you learned to do when you learned to take derivatives. But in this section you will be starting with a pile of pieces lying on the table and you will have to figure out how to put them back together. Each problem of finding the indefinite integral of a function will be a new adventure. You will have to search around for the fit. And you will become efficient at searching out how the pieces fit only if you practice at it. It will be frustrating at first, but if you keep at it you'll get the hang of it.
The good news is that you will not be going off on this adventure unarmed. What you will learn in this section is a box of tools, each of which might be applicable to an integration problem. Sometimes you will have to apply more than one tool, and finding the order in which you apply them will be part of the puzzle. But before you can do that, you must first learn how to use each tool.
Here is a brief look-ahead at the tools you will be learning:
We will cover each of these methods individually, and there will be practice problems where you can apply each method solo. Once you have learned them all, we will go to practice problems where you will apply the methods in duets, trios, and quartets.
Based upon the last section, it should be clear to you that if have a function, f(x), and you take the derivative of its indefinite integral, you get back the original f(x). And from the discussion so far in this section, it should be apparent now that finding a derivative is nearly always easier than finding an indefinite integral. This suggests a way that you can check your answer every time you find the indefinite integral of a function. That is, take the function you got as your answer and find its derivative. If that does not take you back to the original integrand, then you've made a mistake somewhere. You should do this check on every indefinite integral problem that you get through.
The methods we will cover in the next seven sections will put your algebra and trig skills (especially with regard to trig identities) to the test. There is no other way. It is important that whenever you feel unsure about an algebraic manipulation or a trig identity in the development that follows that you take some time to work it through on your own using pencil and paper. Remember that you will be required to reproduce such steps on the exam. If you follow this bit of advice, it is likely that you will emerge from this unit not only knowing how to integrate, but with your algebra and trig skills greatly strengthened as well.
Something you should know: The dirty little secret that they often don't tell first year students is that just because you can write a function, that does not mean you can necessarily write its integral. The functions we will be integrating, along with the ones you will be assigned in class and have to tackle on the exam, are all carefully selected from among those functions whose integrals you can write. But there are some innocently simple functions such as
f(x) = e-x2/2and
sin(x) f(x) =for which there just is no expression made from any finite combination of powers, roots, and elementary functions whose derivative is f(x). That does not mean that the integrals of these functions do not exist. They do. And the integrals of the two examples above are so useful to mathematics that, although you can't write them, they are given names. The integral of the first is called the normal distribution function. The integral of the second is called the sine integral. There are plenty more like them (whose integrals have scary names like incomplete gamma functions and Jacobian elliptic integrals). Math handbooks have tables of the values of these integrals as well as pictures of their graphs and clever ways of approximating them. The properties of these integrals are well studied. But rest assured you will not be asked to integrate functions like these in a first year course.
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