![]() Subtracting these, we get the same result as before. If you are integrating \(\displaystyle \int \frac\right) dx = \ln\left|u\right| – \ln\left|u+1\right| = \ln\left|e^x-1\right| – \ln\left|e^x\right|\). It looks as if you did not understand the point I made in my reply. So it is time to show more details, comparing the two methods and showing the error that can result: In the above question, how can we write e^x/(e^2x – 3e^x + 2)?Īmia shows an understanding of the partial fractions part by stating the first step (for a different problem), but is trying to do it for the given problem directly, without either making the substitution or doing the equivalent implicitly the way Doctor Fenton indicated. We know if we have a proper rational function, when we make decomposition, the degree of numerator will be less than the degree of denominator by 1. With experience, one can skip steps and hold certain parts of a problem in one’s head but this is risky until you are thoroughly familiar with those details you want to skip.Īmia wrote back, showing some work and making the source of the difficulty a little clearer: If you have any questions, please write back and I will try to explain further.Ī lot of what is taught in math classes has this basic purpose: to make a process easier for a newcomer to carry out correctly. So, it is possible to carry out the decomposition and integration without explicitly making the substitution, but that approach offers opportunities to make subtle errors, while making the substitution explicitly and then carrying out the integration in the new variable is more straightforward, in my opinion. You must consider the e^x in the numerator as part of the differential in the substitution du = e^x dx which arises in transforming the integral with the substitution u = e^x. Even if you can see that you can make the decomposition by viewing the expression as a rational expression in e^x, actually making the substitution u = e^x will show that you must decompose the fraction 1/(u^2 – 3u + 2), and not u/(u^2 – 3u + 2). It isn’t essential to substitute a new variable if you can manage the computation in your head, but it is probably easier for most students to make the substitution to clarify what needs to be done. This is a good reason for asking a general question with a specific case in mind. ![]() Is it possible to use partial fractions directly? If yes, why do we use substitution ?Īll three questions have straightforward answers, which are best given by showing work for the example. Why do we use substitution before using partial fractions to compute the integral e^x/ (e^(2x) – 3e^x + 2)? ![]() Can you skip the substitution?Īmia, another user we have known since the days of the old site, asked this last week: Many of the questions we answer are primarily about “how do you solve this problem?”, but at the same time ask deeper, more general questions: “How do you solve problems like this, and are there alternatives?” Today’s question is a good example of this, and raises an interesting point or two.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |