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1 | 1 | <?xml version='1.0' encoding='UTF-8'?> |
2 | | -<pretext xmlns:ns0="http://www.w3.org/2001/XInclude"> |
| 2 | +<pretext> |
3 | 3 | <article> |
4 | 4 | <section xml:id="ps03-section"> |
5 | 5 | <title>Problem Set 3</title> |
6 | 6 | <subtitle>Trigonometric Substitution and Partial Fractions </subtitle> |
7 | | - <ns0:include href="ps01.ptx" xpointer="xpointer(//section[1]/introduction)" set-xml-id="intro-ps03" /> |
| 7 | + <introduction> |
| 8 | + <p> |
| 9 | + This problem set consists not only of problems similar to what you've seen, but |
| 10 | + also of unique problems you may not have seen before. |
| 11 | + The purpose is for you to apply the concepts you've previously |
| 12 | + learned to new, unfamiliar, and usually more interesting situations. |
| 13 | + In some cases, problems connect ideas from multiple learning objectives. |
| 14 | + </p> |
| 15 | + |
| 16 | + <p> Write full, clear solutions to the problems below. Although the final answer is |
| 17 | + important, being able to convey you understand the underlying concepts is more important. |
| 18 | + So show your work and explain your steps. Write complete sentences in English to connect |
| 19 | + your mathematical statements. Do not use the three-dots notation (∴ or ∵) when you mean <q> |
| 20 | + therefore</q> or <q>because</q>. Do not use an arrow (→) when you mean equals (=) or <q>this |
| 21 | + implies</q> or <q>and now the next step is…</q>. Use the worked examples in the textbook, or |
| 22 | + on solution sets to worksheets or previous problem sets, as a guide for how to write your |
| 23 | + solutions. </p> |
| 24 | + |
| 25 | + <p>Written work should be neat and legible. |
| 26 | + Other than being written in longhand, your work should be as clean as something you would |
| 27 | + turn in for one of your writing classes. |
| 28 | + Write in linear order from left to right and top to bottom. |
| 29 | + Leave ample space for graders to write their comments. |
| 30 | + (Do not upload the exercise sheet with your answers written in the margins.) |
| 31 | + Setting aside one page per exercise is a good rule of thumb. |
| 32 | + </p> |
| 33 | + |
| 34 | + <p> |
| 35 | + Prepare a single PDF and submit it to Gradescope. |
| 36 | + Mark the page of each exercise when you upload your pset. |
| 37 | + Make sure each page is in the proper orientation. |
| 38 | + </p> |
| 39 | + |
| 40 | + </introduction> |
8 | 41 | <exercises> |
9 | 42 | <exercise xml:id="ex-trig-sub-definite-x-squared-plus-four"> |
10 | 43 | <statement> |
11 | | - <p> |
12 | | - Evaluate the definite integral <me>\int_1^3 \frac{dx}{x^2\sqrt{x^2+4}}</me>. Explain your steps and justify your choices for trigonometric substitution. |
13 | | - </p> |
| 44 | + <p> Evaluate the definite integral <me>\int_1^3 \frac{dx}{x^2\sqrt{x^2+4}}</me>. Explain |
| 45 | + your steps and justify your choices for trigonometric substitution. </p> |
14 | 46 | </statement> |
15 | 47 | </exercise> |
16 | 48 |
|
17 | 49 | <exercise xml:id="ex-trig-sub-completing-the-square"> |
18 | 50 | <statement> |
19 | | - <p> |
20 | | - Evaluate the indefinite integral <me>\int \frac{x^2}{\sqrt{4x-x^2}} \,dx</me>. |
21 | | - </p> |
| 51 | + <p> Evaluate the indefinite integral <me>\int \frac{x^2}{\sqrt{4x-x^2}} \,dx</me>. </p> |
22 | 52 | </statement> |
23 | 53 | </exercise> |
24 | 54 |
|
25 | 55 | <exercise xml:id="ex-partial-fractions-long-division"> |
26 | 56 | <statement> |
27 | | - <p> |
28 | | - Evaluate the indefinite integral <me>\int \frac{x^3-4x-10}{x^2-x-6} \,dx</me>. |
29 | | - </p> |
| 57 | + <p> Evaluate the indefinite integral <me>\int \frac{x^3-4x-10}{x^2-x-6} \,dx</me>. </p> |
30 | 58 | </statement> |
31 | 59 | </exercise> |
32 | | - |
33 | | - <exercise xml:id="ex-partial-fractions-repeated-quadratic"> |
| 60 | + |
| 61 | + <exercise xml:id="ex-trig-sub-one-over-x-sqrt"> |
34 | 62 | <statement> |
35 | | - <p> |
36 | | - Evaluate the indefinite integral <me>\int \frac{x}{(x-1)^2(x^2+1)} \,dx</me>. |
37 | | - </p> |
| 63 | + <p> Evaluate the indefinite integral <me>\int \frac{1}{x\sqrt{4-x^2}} \,dx</me>. </p> |
38 | 64 | </statement> |
39 | 65 | </exercise> |
40 | | - |
41 | | - <exercise xml:id="ex-trig-sub-one-over-x-sqrt"> |
| 66 | + <exercise xml:id="ex-partial-fractions-repeated-quadratic"> |
42 | 67 | <statement> |
43 | | - <p> |
44 | | - Evaluate the indefinite integral <me>\int \frac{1}{x\sqrt{4-x^2}} \,dx</me>. |
45 | | - </p> |
| 68 | + <p> Evaluate the definite integral <me>\int_{0}^{1/\sqrt{3}} \frac{x}{(x-1)^2(x^2+1)} |
| 69 | + \,dx</me>. </p> |
46 | 70 | </statement> |
47 | 71 | </exercise> |
48 | 72 |
|
49 | 73 | <exercise xml:id="ex-partial-fractions-u-sub"> |
50 | 74 | <statement> |
51 | | - <p> |
52 | | - Evaluate the indefinite integral <me>\int \frac{1}{e^{2x}-e^x-2} \,dx</me>. |
53 | | - </p> |
| 75 | + <p> Evaluate the indefinite integral <me>\int \frac{1}{e^{2x}-e^x-2} \,dx</me>. </p> |
54 | 76 | </statement> |
55 | 77 | </exercise> |
56 | 78 | </exercises> |
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