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Integration for Calculus, Analysis, and Differential Equations最新文献

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FRONT MATTER 前页
Integration for Calculus, Analysis, and Differential Equations Pub Date : 2018-07-11 DOI: 10.1142/9789813272040_fmatter
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引用次数: 0
Method of Integration by Parts 分部积分法
Integration for Calculus, Analysis, and Differential Equations Pub Date : 2018-07-11 DOI: 10.1142/9789813272040_0004
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引用次数: 0
BACK MATTER 回到问题
Integration for Calculus, Analysis, and Differential Equations Pub Date : 2018-07-11 DOI: 10.1142/9789813272040_bmatter
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引用次数: 0
Trigonometric Substitutions 三角函数替换
Integration for Calculus, Analysis, and Differential Equations Pub Date : 2018-07-11 DOI: 10.1142/9789813272040_0006
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引用次数: 0
Mixed Integration Problems 混合集成问题
Integration for Calculus, Analysis, and Differential Equations Pub Date : 2018-07-11 DOI: 10.1142/9789813272040_0010
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引用次数: 0
Method of Substitution 代换法
Integration for Calculus, Analysis, and Differential Equations Pub Date : 2018-07-11 DOI: 10.1142/9789813272040_0003
{"title":"Method of Substitution","authors":"","doi":"10.1142/9789813272040_0003","DOIUrl":"https://doi.org/10.1142/9789813272040_0003","url":null,"abstract":"When a system of equations is graphed, the solutions are the points where the graphs of the equations intersect. If the graphs never intersect, such as parallel lines, the system has no solution because there are no intersection points. If the graphs are the same, the system has infinitely many solutions because the graphs intersect at every point. We will not cover how to solve systems of equations graphically in this class, but thinking about the solutions from a graphical standpoint can help to make more sense of them. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2","PeriodicalId":424539,"journal":{"name":"Integration for Calculus, Analysis, and Differential Equations","volume":"159 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134005972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Rationalizing Substitutions 合理化替换
Integration for Calculus, Analysis, and Differential Equations Pub Date : 2018-07-11 DOI: 10.1142/9789813272040_0008
A. Mingarelli
{"title":"Rationalizing Substitutions","authors":"A. Mingarelli","doi":"10.1142/9789813272040_0008","DOIUrl":"https://doi.org/10.1142/9789813272040_0008","url":null,"abstract":"In this chapter we look at a few more substitutions that can be used effectively to transform some types of integrals to those involving rational functions. In this way we may be able to integrate the original functions by referring to the method of Partial Fractions from Chapter 8. Before we start though we need to remind the reader of a notion called the least common multiple of two given positive integers. As the phrase suggests, the least common multiple (abbr. lcm) of two numbers x, y (assumed integers) is the smallest number that is a multiple of each one of x and y. For example, the lcm{2, 4} = 4, since 4 is the smallest number that is a multiple of both 2 and itself. Other examples include, the lcm{2, 3, 4} = 12, lcm{2, 3} = 6, lcm{2, 5} = 10, lcm{2, 4, 6} = 12 etc. Thus, given two fractions, say 1/2 and 1/3, the least common multiple of their denominators is 6.","PeriodicalId":424539,"journal":{"name":"Integration for Calculus, Analysis, and Differential Equations","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126851394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Integration of Rational Functions 有理函数的积分
Integration for Calculus, Analysis, and Differential Equations Pub Date : 2018-07-11 DOI: 10.1142/9789813272040_0007
{"title":"Integration of Rational Functions","authors":"","doi":"10.1142/9789813272040_0007","DOIUrl":"https://doi.org/10.1142/9789813272040_0007","url":null,"abstract":"In this section we will take a more detailed look at the use of partial fraction decomposi-tions in evaluating integrals of rational functions, a technique we first encountered in the inhibited growth model example in the previous section. However, we will not be able to complete the story until after the introduction of the inverse tangent function in Section 6.5. We begin with a few examples to illustrate how some integration problems involving rational functions may be simplified either by a long division or by a simple substitution. Example To evaluate x 2 x + 1 dx, we first perform a long division of x + 1 into x 2 to obtain x 2 x + 1 = x − 1 + 1 x + 1. Then x 2 x + 1 dx = x − 1 + 1 x + 1 dx = 1 2 x 2 − x + log |x + 1| + c. Example To evaluate 2x + 1 x 2 + x dx, we make the substitution u = x 2 + x du = (2x + 1)dx. Then 2x + 1 x 2 + x dx = 1 u du = log |u| + c = log |x 2 + x| + c. Example To evaluate x x + 1 dx, we perform a long division of x + 1 into x to obtain x x + 1 = 1 − 1 x + 1. Then x x + 1 dx = 1 − 1 x + 1 dx = x − log |x + 1| + c. Alternatively, we could evaluate this integral with the substitution u = x + 1 du = dx.","PeriodicalId":424539,"journal":{"name":"Integration for Calculus, Analysis, and Differential Equations","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128769143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Direct Integration 直接集成
Integration for Calculus, Analysis, and Differential Equations Pub Date : 2018-07-11 DOI: 10.1142/9789813272040_0002
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引用次数: 0
Improper Integrals 反常积分
Integration for Calculus, Analysis, and Differential Equations Pub Date : 2018-07-11 DOI: 10.1142/9789813272040_0009
Ravi P. Agarwal, Peter Cristina, O. Donal
{"title":"Improper Integrals","authors":"Ravi P. Agarwal, Peter Cristina, O. Donal","doi":"10.1142/9789813272040_0009","DOIUrl":"https://doi.org/10.1142/9789813272040_0009","url":null,"abstract":"If we had no intuition regarding what the definite integral should be (the area under the curve), we might very well move on not realizing we have made a severe miscalculation. However, we do have some intuition, and we call tell by the looking at the graph of f(x) = 1/x2 that there is no conceivable way the area bound x = −1 and x = 1 could be negative, let alone attain the value −2 (see Figure 1). But what have we done wrong? This example is subtly different than the previous examples we have seen in that there is a discontinuity at x = 0. It turns out that this is exactly the problem. Our next question is how we might approach rectifying this problem. We notice that for any interval wholly to the left of x = 0 (say, x = −1 to x = a where a < 0), we can apply the standard formula, and the same applies for any interval wholly to the right of x = 0 as well (x = b to x = 1 where b > 0). We are allowed to break integrals into parts like this, so we will consider one integral on the right and one of the left. But how can we manipulate the left and right integrals to capture the whole area we are looking for? In fact, the answer is perhaps the simplest tool we have available to us: we simply","PeriodicalId":424539,"journal":{"name":"Integration for Calculus, Analysis, and Differential Equations","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121603869","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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