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<?xml version='1.0' encoding='UTF-8'?>
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<pretext xmlns:ns0="http://www.w3.org/2001/XInclude">
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<pretext>
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<article>
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<section xml:id="ps03-section">
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<title>Problem Set 3</title>
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<subtitle>Trigonometric Substitution and Partial Fractions </subtitle>
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<ns0:include href="ps01.ptx" xpointer="xpointer(//section[1]/introduction)" set-xml-id="intro-ps03" />
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<introduction>
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<p>
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This problem set consists not only of problems similar to what you've seen, but
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also of unique problems you may not have seen before.
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The purpose is for you to apply the concepts you've previously
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learned to new, unfamiliar, and usually more interesting situations.
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In some cases, problems connect ideas from multiple learning objectives.
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</p>
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<p> Write full, clear solutions to the problems below. Although the final answer is
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important, being able to convey you understand the underlying concepts is more important.
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So show your work and explain your steps. Write complete sentences in English to connect
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your mathematical statements. Do not use the three-dots notation (∴ or ∵) when you mean <q>
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therefore</q> or <q>because</q>. Do not use an arrow (→) when you mean equals (=) or <q>this
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implies</q> or <q>and now the next step is…</q>. Use the worked examples in the textbook, or
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on solution sets to worksheets or previous problem sets, as a guide for how to write your
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solutions. </p>
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<p>Written work should be neat and legible.
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Other than being written in longhand, your work should be as clean as something you would
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turn in for one of your writing classes.
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Write in linear order from left to right and top to bottom.
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Leave ample space for graders to write their comments.
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(Do not upload the exercise sheet with your answers written in the margins.)
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Setting aside one page per exercise is a good rule of thumb.
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</p>
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<p>
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Prepare a single PDF and submit it to Gradescope.
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Mark the page of each exercise when you upload your pset.
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Make sure each page is in the proper orientation.
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</p>
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</introduction>
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<exercises>
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<exercise xml:id="ex-trig-sub-definite-x-squared-plus-four">
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<statement>
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<p>
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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.
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</p>
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<p> Evaluate the definite integral <me>\int_1^3 \frac{dx}{x^2\sqrt{x^2+4}}</me>. Explain
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your steps and justify your choices for trigonometric substitution. </p>
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</statement>
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</exercise>
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<exercise xml:id="ex-trig-sub-completing-the-square">
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<statement>
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<p>
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Evaluate the indefinite integral <me>\int \frac{x^2}{\sqrt{4x-x^2}} \,dx</me>.
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</p>
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<p> Evaluate the indefinite integral <me>\int \frac{x^2}{\sqrt{4x-x^2}} \,dx</me>. </p>
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</statement>
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</exercise>
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<exercise xml:id="ex-partial-fractions-long-division">
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<statement>
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<p>
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Evaluate the indefinite integral <me>\int \frac{x^3-4x-10}{x^2-x-6} \,dx</me>.
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</p>
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<p> Evaluate the indefinite integral <me>\int \frac{x^3-4x-10}{x^2-x-6} \,dx</me>. </p>
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</statement>
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</exercise>
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<exercise xml:id="ex-partial-fractions-repeated-quadratic">
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<exercise xml:id="ex-trig-sub-one-over-x-sqrt">
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<statement>
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<p>
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Evaluate the indefinite integral <me>\int \frac{x}{(x-1)^2(x^2+1)} \,dx</me>.
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</p>
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<p> Evaluate the indefinite integral <me>\int \frac{1}{x\sqrt{4-x^2}} \,dx</me>. </p>
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</statement>
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</exercise>
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<exercise xml:id="ex-trig-sub-one-over-x-sqrt">
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<exercise xml:id="ex-partial-fractions-repeated-quadratic">
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<statement>
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<p>
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Evaluate the indefinite integral <me>\int \frac{1}{x\sqrt{4-x^2}} \,dx</me>.
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</p>
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<p> Evaluate the definite integral <me>\int_{0}^{1/\sqrt{3}} \frac{x}{(x-1)^2(x^2+1)}
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\,dx</me>. </p>
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</statement>
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</exercise>
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<exercise xml:id="ex-partial-fractions-u-sub">
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<statement>
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<p>
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Evaluate the indefinite integral <me>\int \frac{1}{e^{2x}-e^x-2} \,dx</me>.
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</p>
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<p> Evaluate the indefinite integral <me>\int \frac{1}{e^{2x}-e^x-2} \,dx</me>. </p>
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</statement>
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</exercise>
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</exercises>

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