Section 11: Methods of Integration This section is still under construction

11.1 Solving the Puzzle -- Intro to Methods of Integration

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 pull 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:

 Knowing the antiderivative of the function in advance. Sounds tautological, but there is some wisdom here. You already know that taking the derivative of x to a power results in x to a different power times some constant. You also know that the derivative of sin(x) is cos(x), the derivative of cos(x) is -sin(x), and that ex is its own derivative. It is not difficult to run these transformations in reverse to find antiderivatives of functions like these. Applying the chain rule in reverse. Sometimes you can see exactly how the chain rule would be applied to some function to result in the function whose integral you are trying to find. And there are often clues on just how and when to do this. This method is called simple substitution. Applying the product rule in reverse. It is not always clear when this will be fruitful, but it is often worth trying. Sometimes it will work, sometimes not. When it does, it is called integration by parts. Observing how one part of the integrand is related to another in the same way as a common trigonometric function is related to its own derivative. When this happens you can often write an equivalent integral in terms of the trig function, which might turn out to be easier to integrate than the original. This method is called trig substitution. Algebraically (or by use of identities) munging the integrand into something to which you can apply any one of the preceding methods. This includes breaking the integrand into a sum of simpler functions. And happily there is a sum rule for integrals: The integral of the sum is equal to the sum of the integrals. One of the variations of this method is called integration by partial fractions.

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 most often 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.

FYI

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-x2/2
and
sin(x)
f(x)  =
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.