"Correct" methods lead swiftly towards desired results. "Incorrect" methods usually lead to mistakes and blind alleys. Just occasionally though, following through the consequences of an error can develop one's understanding, and even lead to new methods of doing things. Before tackling the problems at the end, you should work through the following examples, making sure you understand where each term comes from.
Integrating by parts the "wrong" way.
We use the method of integration by parts, when we have an expression which is the product of two terms. The technique involves integrating one term, and differentiating the other, as in the following example:
(1)
If, contrary to normal practice, we integrate and differentiate , we get the following rather nasty looking result:
(2)
If we equate the results (1) and (2), we get the following expression:
(3)
Result (3) gives us an identical expression to the power seriesfor sin.
(Hint: Start with .)
Discussion 1
We have now seen that integrating by parts the "wrong" way does not need to be a mistake, as it can be a means of deriving a power series. However, in the example and in Problem 1, the limits of the definite integral were such that the result was relatively simple. The question is: can we use this technique under more general circumstances to obtain useful results? The next two examples illustrate both the potential and the pitfalls of this approach.
Example 2: Consider the following integral::
Expanding by both methods, we get:
Dividing through by , and rearranging, we get the standard power series:
If you change the limits of the above integral to 0 and x, and use the dummy variable w, you should be able to verify that:
So, it turns out that our "wrong" method can be used to generate a standard power series, though not perhaps the one that the original integral might have suggested. The next example shows that things can get even more complicated.
Example 3: Consider the integral: . (Y)
Integrating, and assuming that the differentials exist and are well behaved,
we get:
If you expand this expression , it turns out to be not particularly pleasant.
Discussion 2
We have seen that integration by parts is a potentially useful method of developing power series. However, so far, it has not resulted in the series one might hope for. In example 2, one could have reasonably hoped to derive the power series for , but we ended up with instead. In example 3, one might have hoped to find a general method for deriving power series for , but the result was far too complicated to be useful. Fortunately, there are ways around these problems, some more elegant than others.
The first thing one can do is experiment with different limits, as in this variation on the theme of example 2:
Example 2b: Consider the integral::Expanding, we get:
You should find, with a little manipulation, that this gives the standard power series for .
Although a step forward, this manipulation of the limits is still rather cumbersome. However, there is another way to achieve the same result. When integrating between limits, it is legitimate to use any primitive of ^{1}, so not only , but also , where is an arbitrary constant will do, as in the following, further development of example 2:
Example 2c: Consider the integral::Expanding, we get:
Now we come to the cunning bit: as can take any value we wish, if we choose a suitable value, we can make half of the definite integral vanish, by ensuring that all the powers of are zero! Since we are using as a dummy variable, we can treat as a constant, and hence put into the above expression. You should be able to verify that this gives the following result:
This is the standard Taylor's series expansion for about . By putting = 0, one obtains the Maclaurin series for .
Problems
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