Problem 57993. Easy Sequences 110: Integration of the Sum of a Recursive Trigonometric Function

Created by Ramon Villamangca

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{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>A trigonometric function, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\text{T}(x)</w:t></w:r></w:customXml><w:r><w:t>, is defined as follows:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\text{T}(x) = \\sin(\\arctan(x))</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x</w:t></w:r></w:customXml><w:r><w:t> in radians</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Applying </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\text{T}(x)</w:t></w:r></w:customXml><w:r><w:t> recursively we define another function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\text{R}(x,n)</w:t></w:r></w:customXml><w:r><w:t>, for integer </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>\\text{R}(x,n)=\\underbrace{\\text{T}(\\text{T}(\\text{T}(...\\text{T}(x))))}\\\\_{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{n times}}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>We then define </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\text{S}(x,n)</w:t></w:r></w:customXml><w:r><w:t> as the sum of value of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\text{R}</w:t></w:r></w:customXml><w:r><w:t> from </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>1</w:t></w:r></w:customXml><w:r><w:t> to </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>\\text{S}(x,n) = \\sum_{k=1}^{n} \\text{R}(x,k) </w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Finally, </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>we are asked to evaluate the integral of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\text{S}</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t> with respect to </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, over the real range </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>[x_1,x_2]</w:t></w:r></w:customXml><w:r><w:t>:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>a=\\text{A}(n,x_1,x_2) = \\int^{x_2}_{x_1} \\text{S}(x,n)\\ \\mathrm{d}x}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>For example for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n=3</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x_1=\\pi</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x_2=2\\pi</w:t></w:r></w:customXml><w:r><w:t>, we have:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[  >> a = integral(@(x) sin(atan(x))+sin(atan(sin(atan(x))))+sin(atan(sin(atan(sin(atan(x)))))),pi,2*pi)\n       a = 7.05797686912156]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Please present the final output rounded-off to 6 decimal places. Therefore the final answer is </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>a=7.057977</w:t></w:r></w:customXml><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>-------------------------</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>NOTE: </w:t></w:r><w:r><w:t>There are a number of ways to do </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/help/matlab/numerical-integration-and-differentiation.html\"><w:r><w:rPr><w:b/></w:rPr><w:t>numerical Integration</w:t></w:r></w:hyperlink><w:r><w:t> in Matlab.  Just make sure that the output would be accurate within 6 decimal places of the value obtained using the </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>integral</w:t></w:r><w:r><w:t> function shown above.</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; "><div style="block-size: 507px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 462.578125px 253.5px; transform-origin: 462.578125px 253.5px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">A trigonometric function, </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="31.5" height="19" style="width: 31.5px; height: 19px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, is defined as follows:</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="137.5" height="19" style="width: 137.5px; height: 19px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">x</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> in radians</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Applying </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD8AAAAmCAYAAABzhkOMAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAAP6ADAAQAAAABAAAAJgAAAADyFT37AAAEHElEQVRoBe2YWYhOYRjHxzb2fRuyZi03lJLCBRqFcWEpDZEQJReWuxG5kISSubDFlRsllKUkM1nChaUpW7LUZCt7ZN9+f877dead95zvnOM7QzlP/b/vfZ/n+T/v9rzL9xUVZZLNQDYD2QxkM5DqDJQT/T5oW6BWrhJnddxYTS1CCfXeli5u9S2E2yGkZdgqwWIg30LISoKcBD3AqqQBKyD++ENcCml8FLbPYEOIT1LTXIjfweyoAeyVb+0jnqVcDV6BD2AWmAgkGsBy0Ax0BMPAdFAM/DGo5kR+B8BxsCanLVxhP6GGgD2gBtwCsWQb3lp5V+ps9Wyy3wG2jEShSblnG7z6Eb6/gAEB9kKoWxDkMbgJ2sQNqFnTQdTIQcw3eFF2gacObjk6TZripy3KSLW1Pl9DjS2HVtRPAZGTyAlILR3EhZ4ujb1uN7cbxXMwH7gW0fbP1YdS6pmr1S1EWXmlndLfL7o9voHrfmXKZZ0tWsDxYe3YK68rSnsmqXyEeNkiz6Gudk5b+rCqsqcMrAUVwN6/k9BtBtOAS6o85QKXMYkuysq74p5BqVUI6qjN2YRC97+yRTxhCzDi78c7lNqqtgxGId4j25C07m/0TowgD/BVRwZG5KjjWuleQIMT11xbKzzdPr51MOt8con2+legCbSvc5d/Xl2SwasTn4AGoHs+rhyGIK5QCrStlPJR5BlO4vUJcrb3fJBfUn1XiMVAK/A6QZCzPo6er5VA31HkheekDHJK2oPv4bWqgWsV4soFH6GW8jpfPV/RTHZgxqU9eO1Zia7AJHINklJdcgW8/1WK9mHeG9r7Tkl78Oa01Xu/ubMH4UodVmYCR4e71rN29jQP61k8RdqD16rp4JGYzvyuRfvUFdfFcy3hO+qNIYrh/bXBqxOm8aCXo3xcMhXlErDDZxzrK4cVtc+VacqaN0GOcVbev2/1UzaqmDs6ascVtzvYC7TyeuGZw9LEUPs7QSfgknGessZlTKI7BEmdEPQCawKiyGScxDka4twU2xQwCGiSz4OLQHqJBqEYT4B0GngV0DvCJdtRyn+pyxhHp/QpA9q/CmiwgXIHkE80SY+B0s8MxuboV5+Jq4nVOdHX57TRZ1cs3eF+u8/1V/EGn59AUGbY/s66/p35DkzHXN8Hncy6Sr3XxZ1XV52r6V8iE/slZftk749Og5GP/lky6U+xnoxBIz9l6j8h/eiFVv4uCFr9UdhKgeuHCuqibkAZqFdjmFRjVJaOCHNqaNsMGtSKLEqx4QleG8tSbCNx6K0wtaeHJ44QTNRVWguibMPgKClalPLnwCMQ+IMjQftt4FwF90D7BPwGo+jlpR8sur50ExRCjhFE/zEMLESwtGMU08BMEOeRFdYnnSftwhwyWzYD2QxkM/Bfz8BPP8zkcVJh+/8AAAAASUVORK5CYII=" width="31.5" height="19" style="width: 31.5px; height: 19px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> recursively we define another function </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="46" height="19" style="width: 46px; height: 19px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, for integer </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">:</span></span></div><div style="block-size: 51px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 25.5px; text-align: left; transform-origin: 384px 25.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                </span></span><span style="vertical-align:-20px"><img src="data:image/png;base64,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" width="166.5" height="51" style="width: 166.5px; height: 51px;"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">We then define </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="44" height="19" style="width: 44px; height: 19px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> as the sum of value of </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);">R</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> from </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);">1</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> to </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">:</span></span></div><div style="block-size: 46px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 23px; text-align: left; transform-origin: 384px 23px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                </span></span><span style="vertical-align:-17px"><img src="data:image/png;base64,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" width="129.5" height="46" style="width: 129.5px; height: 46px;"></span></div><div style="block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Finally, </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">we are asked to evaluate the integral of </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);">S</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; "> with respect to </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">x</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">, over the real range </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="43" height="20" style="width: 43px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">:</span></span></div><div style="block-size: 48px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 24px; text-align: left; transform-origin: 384px 24px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                </span></span><span style="vertical-align:-21px"><img src="data:image/png;base64,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" width="204" height="48" style="width: 204px; height: 48px;"></span></div><div style="block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">For example for </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEkAAAAkCAYAAADFGRdYAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAASaADAAQAAAABAAAAJAAAAABLVRfgAAADIElEQVRoBe2YW4hNURjHz8glMjR4MOX2Jinx4IEXuWvyIkXxMkpKCXlQLqVEpCaePEx5kFIoeSO3TiglyjXhiaFIcr9ff/9xvvrObu05e5+hmXNmffWbvfb3fWvP+v5n7bXWOYVCtKhAVCAqEBWICtSbAg31VlCFekYSb4HJMAyuw1W4D9FQYBW8gt8BDuDraxOGksttBbcS5wMcgzPwBrxgO7jvs9ZM5Z/gLIxyKgykvRFMqNsu1ueaO6n4GgxKqfwefhOqMZTTL+SsM5/EaYWvKXW9KPm1Xul1jJZQQAJ+BM2ko4lYplu9v/NhMyxxPcbTXgf7Ya7z12JzKYOWQM9hdJ4C2kh+Cvae6rqy9ABto5qSPja9FKu1i85LWtD1mo3NO/iJdFgG30BifIcmWA8/QDPoNJhQa2lntZMkalDdpTszWDNmD1h9quMSjINcNoDs96AHXAS9blJ9Acg2gIm0qNOT7c8518/6V3PN8z/9yLRM6EMP/c8O/EN9srX7WyNxncG9ddD54TCsAZ01ZFP+Xjr/3nLtSs29JJyolJQhfiNDTijlEE69Bfp6Mhu2g9U5hvYW2AaZbBdZpvZj2u2JXg9K8bsJf63dTmDA58FqfZinAB2+fMchrrPeXYu1OX+tNhsZuK1PWnN1Eq9omoo/wYSYk+ix2sUWJmK1elt0Nan+MgutSfPIsJO41iAt3N50dpJ9Ae0KeUy726w8HVJytfteSIlV475MJ43rNWjnLbOQSLaDKVE/IXiTeLb96sGffTBDW1N7RIa8Sinaff+l2ey5E3poVyK9pIO2bG/TuLEHapbJtGMcgSJUsp7e3ULjk+A2MZKTIpRfmITX1qL2QIb/aUGC7YZHMDyQ21tcyxnIKdD2PjgwqK34VHMxEAu6vAiLAxn78JmIet3egj8zBbr0uKuDEdiYn9BuhWbQh3wQFNO6q6Ugkx0nS530zTikegv+X6WcZ1xnQm+3TQzQRPLXd/ivgDaBBki1LoMpvexH9JvE8y7cKY/87259H50KTaCZpUOwZpVEixYViApEBaICUYGoQFQgKlClAn8AYTTWbkfCa8gAAAAASUVORK5CYII=" width="36.5" height="18" style="width: 36.5px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="42.5" height="20" style="width: 42.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGQAAAAoCAYAAAAIeF9DAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAAZKADAAQAAAABAAAAKAAAAACyuGsJAAAEwklEQVRoBe2Ya6gVVRSATb1a9MeyrCj6cUMtLIv+lEWBJgY96UcKIWmE9aMQI4x+GFekMnobUVHRNcgoAskgi8qKHhglkoJJaS8rMETTQulhj++r2TBMs8+Z25lzOkf3gu/uNWuvWbNm7cfsc4cNS5IqkCqQKpAqkCqQKpAqkCpQfwUOqT9kT0fsI/upMBkmwJewFj6EfZCkgxU4nWd9DH+WsBPbWZCkQxUYz3NcAWWDEWw/0T+tQ/kc1I9x234HfoelcByMgJOy619pw6CsQ0/S5gpMIb4FXxR5zrysXx8H7YiIXzLXVIE7iLMZhkfiuYK2QFgll0X8ajHHkqgleI8EGUOei+GPSL4OxIZcX8wv5/Lf1ZENbp1I33Toh1XgPhtkLMq14PK9C/ZAr8oNFRL/OefzRU4vU53kx8LR0OhnxXb6pak4EC7h/RCWqQk5CIr930HoG9B4gMt7vJ/v+2mD95xE3zPwC4TaNGr1rSSH4+WPIkf6AQhBZ6EfBZ+Dq2U1fA8XQ13iWd8zf6u8VldCxBkHv4F1WABlMgOjx+JQqyrtnLJAzWz+WArBl6GvhLegD9oh5xI0PK+V9oMak1uY5bSBtuy9nbx7wQm6GELe96NfA/Mzm8dnt3lt4nb/L2m0x+ls/y7ww6fshlPBLasd4nNm1hB4GzFerSGOv0ncvkeDk2U9FMXt7Fl4BE6BT8Di+w35EebCIKyB6dCyvEKEMOo3thytdwI4GVeCp6orG6Q9Ktd3H7q1ejlneyOz3ZSztaTelgX0QZMbRHLp3gMu3SdhNjRbgbh0rSwhM995YcUMHZgd2T3zsnuOp/XHpHGsTy1iQgaU6yMRL8S+L/MJvrZPgYeDXpOrSdj8lw4h8ZnZPa4otzrlFjDOVi/qEP+fkz89lB3VDsPnW3B1OAvOhAchDMx56FWlG05Zl5Osp6pHqyad+Xmy8539V32QjSjaHg6GVtqR3LwW3gVPGAb+GopyCYbni0aunwPvGSjpi5n+71PWBSTmb64VEFvZno5uh7ycxoUrw/e9NeuYlF1ruzSzjaN1G4uKRY+J347xcAbcDH4/TszwFHMRnAAuR32L8hGGWeAKqyqbcIxti1Vj6Gd+Q5WzuWEVONPngAUuSh+G5eD2HOQYlMfB76UraxCUGf80f//9jL/2PwEvQvBBjUs/XRb5UHDZ7ocwslegO9LiScvj3U6YCjExSf0d0G4XJ9suMF9X+/ISXsLmO+szDZS58ANoE3eFII+hBPvb6G/CV+CgVpJv8DLAXnB23A1BxqA4K0L/bvR7Q2ek3YzdPdSZ0c3id287hOI1a90RfKfRsCd3n3VzmwpyJ0ox1pBWv9tFCLACfXiInLVhxuvjLBpR6M9fOoPcqk7OG7tUf528wntXacM3wp1kR+7eqwrvdw7X4chru6TQ3/RyLB5uURMjns6K82FKpD+Y/ehtgdnBcAC3bnUDkP9e5F/Xrd3PwJF5Yyf1UTzMvXLIs6GTSR4sz3ILewEeKnlhl3eSDlbA7WwQngb1vFzHxaK8IenxChSLF/ds3LOM7vnwPnhcDuIPoQnQD9uCMbXtrYBHuUank9XtfXyKniqQKpAqkCqQKpAqkCqQKpAqkCrQYxX4CzYWfmZMLQJmAAAAAElFTkSuQmCC" width="50" height="20" style="width: 50px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, we have:</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 40px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 459.578125px 20px; transform-origin: 459.578125px 20px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 459.578125px 10px; transform-origin: 459.578125px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">  &gt;&gt; a = integral(@(x) sin(atan(x))+sin(atan(sin(atan(x))))+sin(atan(sin(atan(sin(atan(x)))))),pi,2*pi)</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 459.578125px 10px; transform-origin: 459.578125px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">       a = 7.05797686912156</span></span></div></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Please present the final output rounded-off to 6 decimal places. Therefore the final answer is </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="85.5" height="18" style="width: 85.5px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">-------------------------</span></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">NOTE: </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">There are a number of ways to do </span></span><a target='_blank' href = "https://www.mathworks.com/help/matlab/numerical-integration-and-differentiation.html"><span style=""><span style="font-weight: 700; ">numerical Integration</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> in Matlab.  Just make sure that the output would be accurate within 6 decimal places of the value obtained using the </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-weight: 700; ">integral</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> function shown above.</span></span></div></div></div>

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Problem 57993. Easy Sequences 110: Integration of the Sum of a Recursive Trigonometric Function

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