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The lure is that using these procedures is fun for certain people. \ \nGood practitioners of the traditional pat procedures view an integral as a \ battle and they view the traditional pat procedures as strategies to employ \ to win the battle. \nOthers who enjoy the traditional pat procedures are \ those who aspire to write mathematical software like ", StyleBox["Mathematica", FontSlant->"Italic"], ". ", StyleBox["Mathematica", FontSlant->"Italic"], " is a supreme practitioner of the traditional pat procedures. In fact, \ everything ", StyleBox["Mathematica", FontSlant->"Italic"], " does is based on some pat procedure - traditional or otherwise. That \ means that some people at Wolfram Research understand this stuff very deeply.\ \nTry your hand at the traditional pat procedures by using them to attempt to \ calculate the following integrals. 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If you cannot handle this type of calculation by hand, then you are deficient \ in this aspect of calculus.\ \>", "SmallText"] }, Closed]], Cell["T.2.b.i)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Calculate\n ", Cell[BoxData[ \(\[Integral]\_0\%t\ \(x\^3\) Cos[2 x] \[DifferentialD]x\)]], "." }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "Go with the technique of integration by differentiation from the last \ lesson.\nPut\n ", Cell[BoxData[ \(g[s] = Sin[s\ x]\)]], " \n and look at the third derivative of ", Cell[BoxData[ \(g[s]\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, g, s, x];\)\), "\n", \(\(g[s_]\ = \ Sin[s\ x];\)\), "\n", \(\(\(\(g'\)'\)'\)[s]\)}], "Input"], Cell[BoxData[ \(\(-x\^3\)\ Cos[s\ x]\)], "Output"] }, Open ]], Cell[TextData[{ "This tells you that when you go with \n ", Cell[BoxData[ \(f[s] = \[Integral]\_0\%t Sin[s\ x] \[DifferentialD]x\)]], ",\n then you are guaranteed that\n ", Cell[BoxData[ \(\[Integral]\_0\%t\( x\^3\) Cos[2 x] \[DifferentialD]x\)]], " \nis given by the third derivative\n ", Cell[BoxData[ RowBox[{"-", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "2", "]"}]}]]], ".\nTry it:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, s];\)\), "\n", \(\(f[s_]\ = \ Integrate[Sin[s\ x], {x, 0, t}];\)\), "\n", \(handcalculation\ = \ \(-\(\(\(f'\)'\)'\)[2]\)\)}], "Input"], Cell[BoxData[ \(3\/8 - 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\(45\ \[ExponentialE]\^\(\(-4\)\ t\)\)\/1024 - 45\/256\ \[ExponentialE]\^\(\(-4\)\ t\)\ t - 45\/128\ \[ExponentialE]\^\(\(-4\)\ t\)\ t\^2 - 15\/32\ \[ExponentialE]\^\(\(-4\)\ t\)\ t\^3 - 15\/32\ \[ExponentialE]\^\(\(-4\)\ t\)\ t\^4 - 3\/8\ \[ExponentialE]\^\(\(-4\)\ t\)\ t\^5 - 1\/4\ \[ExponentialE]\^\(\(-4\)\ t\)\ t\^6\)], "Output"] }, Open ]], Cell["Check:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(mathematicacalculation\ = \ Integrate[x^6\ \ E^\((\(-4\)\ x)\), {x, 0, t}]\)], "Input"], Cell[BoxData[ \(45\/1024 - \(\[ExponentialE]\^\(\(-4\)\ t\)\ \((45 + 180\ t + 360\ t\^2 \ + 480\ t\^3 + 480\ t\^4 + 384\ t\^5 + 256\ t\^6)\)\)\/1024\)], "Output"] }, Open ]], Cell["Multiply out both:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[handcalculation]\)], "Input"], Cell[BoxData[ \(45\/1024 - \(45\ \[ExponentialE]\^\(\(-4\)\ t\)\)\/1024 - 45\/256\ \[ExponentialE]\^\(\(-4\)\ t\)\ t - 45\/128\ \[ExponentialE]\^\(\(-4\)\ t\)\ t\^2 - 15\/32\ \[ExponentialE]\^\(\(-4\)\ t\)\ t\^3 - 15\/32\ \[ExponentialE]\^\(\(-4\)\ t\)\ t\^4 - 3\/8\ \[ExponentialE]\^\(\(-4\)\ t\)\ t\^5 - 1\/4\ \[ExponentialE]\^\(\(-4\)\ t\)\ t\^6\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[mathematicacalculation\ ]\)], "Input"], Cell[BoxData[ \(45\/1024 - \(45\ \[ExponentialE]\^\(\(-4\)\ t\)\)\/1024 - 45\/256\ \[ExponentialE]\^\(\(-4\)\ t\)\ t - 45\/128\ \[ExponentialE]\^\(\(-4\)\ t\)\ t\^2 - 15\/32\ \[ExponentialE]\^\(\(-4\)\ t\)\ t\^3 - 15\/32\ \[ExponentialE]\^\(\(-4\)\ t\)\ t\^4 - 3\/8\ \[ExponentialE]\^\(\(-4\)\ t\)\ t\^5 - 1\/4\ \[ExponentialE]\^\(\(-4\)\ t\)\ t\^6\)], "Output"] }, Open ]], Cell["Dead ringers.", "SmallText"] }, Closed]], Cell["T.2.c)", "Subsubsection"], Cell[TextData[{ "Here's ", StyleBox["Mathematica", FontSlant->"Italic"], "'s calculation of \n ", Cell[BoxData[ \(\[Integral]\_3.2\%10.5\ f[x] \[DifferentialD]x\)]], "\nfor\n ", Cell[BoxData[ \(f[x] = x\^2\/\((x\ + \ 4)\)\^2\)]], ": " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[f, x];\)\), "\n", \(\(f[x_] = x\^2\/\((x + 4)\)\^3;\)\), "\n", \(\[Integral]\_3.2\%10.5 f[x] \[DifferentialD]x\)}], "Input"], Cell[BoxData[ \(0.2569516973317949`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Explain where this calculation comes from:", "Text"], Cell["Answer by partial fractions:", "Special1"], Cell[TextData[{ "Take another look at ", Cell[BoxData[ \(f[x]\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(f[x]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(x\^2\/\((4 + x)\)\^3\)], "Output"] }, Open ]], Cell["Put:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[newf, a, b, c];\)\), "\n", \(newf[x_] = a\/\(4 + x\) + b\/\((4 + x)\)\^2 + c\/\((4 + x)\)\^3\)}], "Input"], Cell[BoxData[ \(c\/\((4 + x)\)\^3 + b\/\((4 + x)\)\^2 + a\/\(4 + x\)\)], "Output"] }, Open ]], Cell[TextData[{ "Now put ", Cell[BoxData[ \(newf[x]\)]], " over a common denominator:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Together[newf[x]]\)], "Input"], Cell[BoxData[ \(\(16\ a + 4\ b + c + 8\ a\ x + b\ x + a\ x\^2\)\/\((4 + x)\)\^3\)], \ "Output"] }, Open ]], Cell["You want this to be:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(f[x]\)], "Input"], Cell[BoxData[ \(x\^2\/\((4 + x)\)\^3\)], "Output"] }, Open ]], Cell[TextData[{ "You can arrange this by solving for the ", Cell[BoxData[ \(a\)]], ", ", Cell[BoxData[ \(b\)]], " and ", Cell[BoxData[ \(c\)]], " that make:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(eqn1\ = \ \((16\ a\ + 4\ b\ + \ c == \ 0)\);\)\), "\n", \(\(eqn2\ = \ \((8\ a\ + \ b\ == 0)\);\)\), "\n", \(\(eqn3\ = \ \((\ a == \ 1)\);\)\), "\n", \(Solve[{eqn1, eqn2, eqn3}]\)}], "Input"], Cell[BoxData[ \({{a \[Rule] 1, b \[Rule] \(-8\), c \[Rule] 16}}\)], "Output"] }, Open ]], Cell[TextData[{ "This tells you that when you go with ", Cell[BoxData[ \(a = 1\)]], ", ", Cell[BoxData[ \(b = \(-8\)\)]], " and ", Cell[BoxData[ \(c = 16\)]], " then \n ", Cell[BoxData[ \(newf[x] = f[x]\)]], "." }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(a\ = 1;\)\), "\n", \(\(b\ = \ \(-8\);\)\), "\n", \(\(c\ = \ 16;\)\), "\n", \(newf[x]\)}], "Input"], Cell[BoxData[ \(16\/\((4 + x)\)\^3 - 8\/\((4 + x)\)\^2 + 1\/\(4 + x\)\)], "Output"] }, Open ]], Cell["Confirm:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Apart[f[x]]\)], "Input"], Cell[BoxData[ \(16\/\((4 + x)\)\^3 - 8\/\((4 + x)\)\^2 + 1\/\(4 + x\)\)], "Output"] }, Open ]], Cell[TextData[{ "The point of this is that \n ", Cell[BoxData[ \(\[Integral]\_3.2\%10.5\ newf[x] \[DifferentialD]x\)]], "\nis easy to integrate by hand.\nIt is\n ", Cell[BoxData[ RowBox[{\(16\/\(\((\(-2\))\) \((4 + x)\)\^2\) - 8\/\(\((\(-1\))\) \((4 + x)\)\) + \ Log[4\ + \ x]\), SubsuperscriptBox[ StyleBox["|", FontSize->24], "3.2", "10.5"]}]]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(top\ = \((16\/\(\((\(-2\))\) \((4 + x)\)\^2\) - 8\/\(\((\(-1\))\) \((4 + x)\)\) + \ Log[4\ + \ x])\) /. x -> 10.5;\)\), "\n", \(\(bottom\ = \((16\/\(\((\(-2\))\) \((4 + x)\)\^2\) - 8\/\(\((\(-1\))\) \((4 + x)\)\) + \ Log[4\ + \ x])\) /. x -> 3.2;\)\n\), "\n", \(integral\ = \ top\ - \ bottom\)}], "Input"], Cell[BoxData[ \(0.2569516973317958`\)], "Output"] }, Open ]], Cell[TextData[{ "This is equivalent to the ", StyleBox["Mathematica", FontSlant->"Italic"], " calculation:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, t];\)\), "\n", \(\[Integral]\_3.2\%10.5 f[x] \[DifferentialD]x\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.2569516973317949`\)], "Output"] }, Open ]], Cell["Second answer by substitution", "Special1"], Cell[TextData[{ "The integral is ;\n", Cell[BoxData[ \(\(\(\ \ \ \ \ \ \ \)\(\[Integral]\_3.2\%10.5\( x\^2\/\((x\ + \ \ 4)\)\^3\) \[DifferentialD]x\)\)\)]], ".\n\nMake the substitution ", Cell[BoxData[ \(u = x + 4\)]], " (", Cell[BoxData[ \(x = u - 4\)]], ") to get a new equivalent integral\n ", Cell[BoxData[ \(\[Integral]\_\(3.2\ + \ \ 4\)\%\(10.5\ + \ 4\)\(\((u\ - \ \ 4)\)\^2\/u\^3\) \[DifferentialD]u\)]], " = ", Cell[BoxData[ \(\[Integral]\_7.2\%14.5\(\((u\ - \ 4)\)\^2\/u\^3\) \ \[DifferentialD]u\)]], ".\n No multiply ", Cell[BoxData[ \(\((u\ - \ 4)\)\^2\/u\^3\)]], " out:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\((u - 4)\)\^2\/u\^3]\)], "Input"], Cell[BoxData[ \(16\/u\^3 - 8\/u\^2 + 1\/u\)], "Output"] }, Open ]], Cell[TextData[{ "Now you've got the integral in a form ripe for easy calculation:\n", Cell[BoxData[ \(\(\(\ \ \ \ \)\(\[Integral]\_3.2\%10.5\( x\^2\/\((x\ + \ 4)\)\^3\) \ \[DifferentialD]x\)\)\)]], " ", Cell[BoxData[ \( = \)]], " ", Cell[BoxData[ \(\[Integral]\_7.2\%14.5 16\/u\^3 - 8\/u\^2 + \(1\/u\) \(\(\[DifferentialD]\)\(u\)\(\ \)\)\)]], "\n ", Cell[BoxData[ \( = \)]], " ", Cell[BoxData[ RowBox[{" ", RowBox[{\(\(-\(16\/\(2\ u\^2\)\)\) - \(-\(8\/u\)\) + Log[u]\), SubsuperscriptBox[ StyleBox["|", FontSize->24], "7.2", "14.5"], " ", ":"}]}]]] }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(top\ = \ \((\(-\(16\/\(2\ u\^2\)\)\) - \(-\(8\/u\)\) + Log[u])\) /. u -> 14.5;\)\), "\n", \(\(bottom\ = \ \((\(-\(16\/\(2\ u\^2\)\)\) - \(-\(8\/u\)\) + Log[u])\) /. u -> 7.2;\)\), "\n", \(top\ - \ bottom\)}], "Input"], Cell[BoxData[ \(0.2569516973317958`\)], "Output"] }, Open ]], Cell[TextData[{ "This is equivalent to the ", StyleBox["Mathematica", FontSlant->"Italic"], " calculation:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, t];\)\), "\n", \(\[Integral]\_3.2\%10.5 f[x] \[DifferentialD]x\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0.2569516973317949`\)], "Output"] }, Open ]], Cell["And you're out of here.", "SmallText"] }, Closed]], Cell["T.2.d.i)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Calculate\n ", Cell[BoxData[ \(\[Integral]\_0\%1.2\( x\^4\) \(\((4 - x\^2)\)\^\(5\/2\)\) \[DifferentialD]x\)]], "." }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "The integral absorbing your attention is ", Cell[BoxData[ \(\[Integral]\_0\%\(\(1.2\)\(\ \)\)\(x\^4\) \(\((4 - x\^2)\)\^\(5\/2\)\) \[DifferentialD]x\)]], ".\nThere is a definite geek in this integral; it's the term\n ", Cell[BoxData[ \(\((4 - x\^2)\)\^\(5\/2\)\)]], ".\nLook at the following identities:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[t];\)\), "\n", \(TrigExpand[Sin[t]\^2 + Cos[t]\^2]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[t];\)\), "\n", \(TrigExpand[Cosh[t]\^2 - Sinh[t]\^2]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[TextData[{ "The first identity tells you that\n ", Cell[BoxData[ \(Cos[t]\^2 = 1 - Sin[t]\^2\)]], ". \nMultiplying by ", Cell[BoxData[ \(4\)]], " gives\n ", Cell[BoxData[ \(4 Cos[t]\^2 = 4 - 4 Sin[t]\^2\)]], ".\nYou can squash the geek in ", Cell[BoxData[ \(\[Integral]\_0\%1\( x\^4\) \(\((4 - x\^2)\)\^\(5\/2\)\) \[DifferentialD]x\)]], " by setting\n ", Cell[BoxData[ \(x = 2 Sin[t]\)]], ".\nThis substitution gives the pairings\n ", Cell[BoxData[ \(\(x\^4\) \((1 - x\^2)\)\^\(5\/2\)\)]], " <----> ", Cell[BoxData[ \(\(2\^4\) \(Sin[t]\^4\) \((4 - 4 Sin[t]\^2)\)\^\(5\/2\)\)]], "\n ", Cell[BoxData[ \(\(\(=\)\(\(2\^4\) \(Sin[t]\^4\) \(4\^\(5\/2\)\) Cos[t]\^5\)\)\)]], ";\n ", Cell[BoxData[ \(\[DifferentialD]x\)]], " <--------------> ", Cell[BoxData[ \(2 Cos[t] \[DifferentialD]t\)]], ";\n ", Cell[BoxData[ \(\[Integral]\_0\%1.2\)]], " <-------------> ", Cell[BoxData[ RowBox[{\(\[Integral]\_0\), ErrorBox[\(\\(\(\%\)\(ArcSin[0.6]\)\(\ \)\\)\)]}]]], Cell[BoxData[ \(\ \)]], "\nbecause ", Cell[BoxData[ \(t = 0\)]], " when ", Cell[BoxData[ \(x = 0\)]], ":" }], "SmallText"], Cell[TextData[{ " and ", Cell[BoxData[ \(t = ArcSin[0.6]\)]], " when ", Cell[BoxData[ \(x = 1.2\)]], ":" }], "SmallText"], Cell[TextData[{ "This tells you that\n ", Cell[BoxData[ \(\[Integral]\_0\%\(\(1.2\)\(\ \)\)\(x\^4\) \(\((4 - x\^2)\)\^\(5\/2\)\) \[DifferentialD]x\)]], "\n ", Cell[BoxData[ \(\(\(=\)\(\[Integral]\_0\%\(ArcSin[0.6]\)\(2\^4\) \(Sin[ t]\^4\) \(4\^\(5\/2\)\) \(Cos[t]\^5\) 2 Cos[t] \[DifferentialD]t\)\)\)]], ". " }], "SmallText"], Cell["Now apply some trig identities:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[t];\)\), "\n", \(TrigReduce[2\^4\ 4\^\(5/2\)\ 2\ Sin[t]\^4\ Cos[t]\^6]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(2\ \((6 + 2\ Cos[2\ t] - 8\ Cos[4\ t] - 3\ Cos[6\ t] + 2\ Cos[8\ t] + Cos[10\ t])\)\)], "Output"] }, Open ]], Cell[TextData[{ "This tells you that\n ", Cell[BoxData[ \(\[Integral]\_0\%1.2\( x\^4\) \(\((4 - x\^2)\)\^\(5\/2\)\) \[DifferentialD]x\)]], "\n ", Cell[BoxData[ \(\(\(=\)\(\[Integral]\_0\%\(ArcSin[0.6]\)2\ \((6 + 2\ Cos[2\ t] - 8\ Cos[4\ t] - 3\ Cos[6\ t] + 2\ Cos[8\ t] + Cos[10\ t])\) \[DifferentialD]t\)\)\)]], ".\nThis is\n", Cell[BoxData[ \(2\ \((6\ t + 2\/2\ Sin[2\ t] - 8\/4\ Sin[4\ t] - 3\/6\ Sin[6\ t] + 2\/8\ Sin[8\ t] + 1\/10\ Sin[10\ t])\)\)]], Cell[BoxData[ SubsuperscriptBox[ StyleBox["|", FontSize->24], "0", \(ArcSin[0.6]\)]]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(top\ = \ 2\ \((6\ t + 2\/2\ Sin[2\ t] - 8\/4\ Sin[4\ t] - 3\/6\ Sin[6\ t] + 2\/8\ Sin[8\ t] + 1\/10\ Sin[10\ t])\) /. t -> ArcSin[0.6];\)\), "\n", \(\(bottom\ = \ 2\ \((6\ t + 2\/2\ Sin[2\ t] - 8\/4\ Sin[4\ t] - 3\/6\ Sin[6\ t] + 2\/8\ Sin[8\ t] + 1\/10\ Sin[10\ t])\) /. t -> 0;\)\), "\n", \(top\ - \ bottom\)}], "Input"], Cell[BoxData[ \(7.727501617839414`\)], "Output"] }, Open ]], Cell["Check:", "SmallText", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(NIntegrate[x\^4\ \((4 - x\^2)\)\^\(5/2\), {x, 0, 1.2}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(7.72750161783937`\)], "Output"] }, Open ]], Cell["Nailed it.", "SmallText"] }, Closed]], Cell["T.2.d.ii)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Calculate\n ", Cell[BoxData[ \(\[Integral]\_0\%1.2\ \(x\^4\) \(\((4 + x\^2)\)\^\(5\/2\)\) \[DifferentialD]x\)]], "." }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "The integral absorbing your attention is ", Cell[BoxData[ \(\[Integral]\_0\%1.2\( x\^4\) \(\((4 + x\^2)\)\^\(5\/2\)\) \[DifferentialD]x\)]], ".\nThere is a definite geek in this integral; it's the term\n ", Cell[BoxData[ \(\((4 + x\^2)\)\^\(5\/2\)\)]], ".\nLook at the following identities:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[t];\)\), "\n", \(TrigExpand[Sin[t]\^2 + Cos[t]\^2]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[t];\)\), "\n", \(TrigExpand[Cosh[t]\^2 - Sinh[t]\^2]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[TextData[{ "The second identity tells you that\n ", Cell[BoxData[ \(Cosh[t]\^2 = Sinh[t]\^2 + 1\)]], ". \nMultiplying by ", Cell[BoxData[ \(4\)]], " gives\n ", Cell[BoxData[ \(4 Cosh[t]\^2 = 4 + 4 Sinh[t]\^2\)]], ".\nYou can squash the geek in ", Cell[BoxData[ \(\[Integral]\_0\%2\( x\^4\) \(\((4 + x\^2)\)\^\(5\/2\)\) \[DifferentialD]x\)]], " by setting\n ", Cell[BoxData[ \(x = 2 Sinh[t]\)]], ".\nThis substitution gives the pairings\n ", Cell[BoxData[ \(\(x\^4\) \((1 - x\^2)\)\^\(5\/2\)\)]], " <----> ", Cell[BoxData[ \(\(2\^4\) \(Sinh[t]\^4\) \((4 + 4 Sinh[t]\^2)\)\^\(5\/2\)\)]], "\n ", Cell[BoxData[ \(\(\((\(=\)\(2\))\)\^4\) \(Sinh[t]\^4\) \(4\^\(5\/2\)\) Cosh[t]\^5\)]], ";\n ", Cell[BoxData[ \(\[DifferentialD]x\)]], " <--------------> ", Cell[BoxData[ \(2 Cosh[t] \[DifferentialD]t\)]], ";\n ", Cell[BoxData[ \(\[Integral]\_0\%1.2\)]], " <--------------> ", Cell[BoxData[ \(\[Integral]\_0\%\(ArcSinh[0.6]\)\)]], "\nbecause ", Cell[BoxData[ \(t = 0\)]], " when ", Cell[BoxData[ \(x = 0\)]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(2\ Sinh[0]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[TextData[{ "and ", Cell[BoxData[ \(t = ArcSinh[0.6]\)]], " when ", Cell[BoxData[ \(x = 1.2\)]], ":" }], "SmallText"], Cell[TextData[{ "This tells you that\n ", Cell[BoxData[ \(\[Integral]\_0\%1.2\( x\^4\) \(\((4 + x\^2)\)\^\(5\/2\)\) \[DifferentialD]x\)]], "\n ", Cell[BoxData[ \(\(\(=\)\(\[Integral]\_0\%\(ArcSinh[0.6]\)\(2\^4\) \(Sinh[ t]\^4\) \(4\^\(5\/2\)\) \(Cosh[t]\^5\) 2 Cosh[t] \[DifferentialD]t\)\)\)]], " " }], "SmallText"], Cell["Now apply some identities:", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[t];\)\), "\n", \(TrigReduce[2\^4\ 4\^\(5/2\)\ 2\ Sinh[t]\^4\ Cosh[t]\^6]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(2\ \((6 + 2\ Cosh[2\ t] - 8\ Cosh[4\ t] - 3\ Cosh[6\ t] + 2\ Cosh[8\ t] + Cosh[10\ t])\)\)], "Output"] }, Open ]], Cell[TextData[{ "This tells you that \n ", Cell[BoxData[ \(\[Integral]\_0\%1.2\( x\^4\) \(\((4 - x\^2)\)\^\(5\/2\)\) \[DifferentialD]x\)]], "\n ", Cell[BoxData[ \(\(\(=\)\(\[Integral]\_0\%\(ArcSinh[0.6]\)\(2\^4\) \(Sinh[ t]\^4\) \(4\^\(5\/2\)\) \(Cosh[t]\^5\) 2 Cosh[t] \[DifferentialD]t\)\)\)]], " \n ", Cell[BoxData[ \(\(\(=\)\(\[Integral]\_0\%\(ArcSinh[0.6]\)2\ \((6 + 2\ Cosh[2\ t] - 8\ Cosh[4\ t] - 3\ Cosh[6\ t] + 2\ Cosh[8\ t] + Cosh[10\ t])\) \[DifferentialD]t\)\)\)]], ".\nThis is\n", Cell[BoxData[ \(2\ \((6\ t + 2\/2\ Sinh[2\ t] - 8\/4\ Sinh[4\ t] - 3\/6\ Sinh[6\ t] + 2\/8\ Sinh[8\ t] + 1\/10\ Sinh[10\ t])\)\)]], Cell[BoxData[ SubsuperscriptBox[ StyleBox["|", FontSize->24], "0", \(ArcSin[0.6]\)]]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(top\ = \ 2\ \((6\ t + 2\/2\ Sinh[2\ t] - 8\/4\ Sinh[4\ t] - 3\/6\ Sinh[6\ t] + 2\/8\ Sinh[8\ t] + 1\/10\ Sinh[10\ t])\) /. t -> ArcSinh[0.6];\)\), "\n", \(\(bottom\ = \ 2\ \((6\ t + 2\/2\ Sinh[2\ t] - 8\/4\ Sinh[4\ t] - 3\/6\ Sinh[6\ t] + 2\/8\ Sinh[8\ t] + 1\/10\ Sinh[10\ t])\) /. t -> 0;\)\), "\n", \(top\ - \ bottom\)}], "Input"], Cell[BoxData[ \(28.414671351064456`\)], "Output"] }, Open ]], Cell["Check:", "SmallText", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(NIntegrate[x\^4\ \((4 + x\^2)\)\^\(5/2\), {x, 0, 1.2}]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(28.414671351064467`\)], "Output"] }, Open ]], Cell["Nailed it.", "SmallText"] }, Closed]], Cell["T.2.e)", "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ "Calculate\n ", Cell[BoxData[ RowBox[{\(\[Integral]\_1\%16\), RowBox[{ StyleBox[\(1\/\(1 + x\^\(1\/4\)\)\), FontSize->16], \(\[DifferentialD]x\)}]}]]], "." }], "Text"], Cell["Answer:", "Special1"], Cell[TextData[{ "There's a smelly geek in\n ", Cell[BoxData[ RowBox[{\(\[Integral]\_1\%16\), RowBox[{ StyleBox[\(1\/\(1 + x\^\(1\/4\)\)\), FontSize->16], \(\[DifferentialD]x\)}]}]]], "\n\nand it's the ", Cell[BoxData[ \(x\^\(1\/4\)\)]], " term.\nSquash the geek by transforming this integral with the \ substitution\n ", Cell[BoxData[ \(t\^4 = x\)]], ".\nThis gives the pairings\n ", Cell[BoxData[ StyleBox[\(1\/\(1 + x\^\(1\/4\)\)\), FontSize->16]]], " <-------> ", Cell[BoxData[ StyleBox[\(1\/\(1 + t\)\), FontSize->16]]], ";\n ", Cell[BoxData[ \(\[DifferentialD]x\)]], " <---------------> ", Cell[BoxData[ \(4 \( t\^3\) \[DifferentialD]t\)]], ";\n ", Cell[BoxData[ \(\[Integral]\_1\%16\)]], " <---------------> ", Cell[BoxData[ \(\[Integral]\_1\%2\)]], 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RowBox[{\(\[Integral]\_1\%16\), RowBox[{ StyleBox[\(1\/\(1 + x\^\(1\/4\)\)\), FontSize->16], \(\[DifferentialD]x\)}]}]]], "\n ", Cell[BoxData[ RowBox[{"=", RowBox[{\(\[Integral]\_1\%2\), RowBox[{ StyleBox[\(\(4 t\^3\)\/\(1 + t\)\), FontSize->16], \(\[DifferentialD]t\)}]}]}]]], "\n ", Cell[BoxData[ RowBox[{"=", RowBox[{\(\[Integral]\_1\%2\), RowBox[{ RowBox[{"(", RowBox[{"4", "-", \(4 t\), "+", \(4 t\^2\), "-", StyleBox[\(4\/\(1 + t\)\), FontSize->16]}], ")"}], \(\[DifferentialD]t\)}]}]}]]], ".\nThis is\n ", Cell[BoxData[ \(\((4\ t - \(4\ t\^2\)\/2 + \(4\ t\^3\)\/3 - 4\ Log[1 + t]\)\)]], ")", Cell[BoxData[ SubsuperscriptBox[ StyleBox["|", FontSize->24], "1", "2"]]], ":" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(top = 4\ t - \(4\ t\^2\)\/2 + \(4\ t\^3\)\/3 - 4\ Log[1 + t] /. t \[Rule] 2;\)\), "\n", \(\(bottom = 4\ t - \(4\ t\^2\)\/2 + \(4\ t\^3\)\/3 - 4\ Log[1 + t] /. t \[Rule] 1;\)\), "\n", \(top - bottom\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(22\/3 + 4\ 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SubsuperscriptBox[ StyleBox["|", FontSize->24], "0", "t"], RowBox[{"+", RowBox[{"(", StyleBox[\(1\/\@\[Pi]\), FontSize->16]}]}]}]}], ")"}], \(E\^\(-x\^2\)\)}], SubsuperscriptBox[ StyleBox["|", FontSize->24], "0", "t"]}]]], "\n ", Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"=", RowBox[{\(t\ Erf[t]\), "-", "0", "+", RowBox[{"(", StyleBox[\(1\/\@\[Pi]\), FontSize->16]}]}]}], ")"}], \((E\^\(-t\^2\)\)}], "-", "1"}], ")"}]]], "\n ", Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"=", RowBox[{\(t\ Erf[t]\), "+", RowBox[{"(", StyleBox[\(1\/\@\[Pi]\), FontSize->16]}]}]}], ")"}], \((E\^\(-t\^2\)\)}], "-", "1"}], ")"}]]], "\nCheck:" }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[x, t];\)\), "\n", \(\[Integral]\_0\%t Erf[x] \[DifferentialD]x\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(\(-\(1\/\@\[Pi]\)\) + \(\[ExponentialE]\^\(-t\^2\)\ \((1 + \ \[ExponentialE]\^\(t\^2\)\ \@\[Pi]\ t\ Erf[t])\)\)\/\@\[Pi]\)], "Output"] }, Open ]] }, Closed]], 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Cell[StyleData["PrefaceText", "TwoColumn"], CellMargins->{{0, Inherited}, {2, 2}}, LineSpacing->{1, 1}, FontSize->12, FontColor->GrayLevel[0], Background->None] }, Closed]], Cell[StyleData["PrefaceHyperlink"], CellMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1, 1}, LimitsPositioningTokens->{}, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Times", FontSize->10, FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[CellGroupData[{ Cell[StyleData["Text"], ShowClosedCellArea->True, CellMargins->{{15, 10}, {Inherited, Inherited}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, LimitsPositioningTokens->{}, FontFamily->"Times", FontSize->16, FontColor->RGBColor[0, 0, 0.500008]], Cell[StyleData["Text", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, Inherited}}, LineSpacing->{2, 0}, FontSize->16, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Text", "Printout"], CellMargins->{{14, Inherited}, {3, 1}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 2}, FontColor->GrayLevel[0], Background->None], Cell[StyleData["Text", "TwoColumn"], CellMargins->{{14, Inherited}, {3, 1}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 2}, FontSize->14, FontColor->GrayLevel[0], Background->None] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], ShowClosedCellArea->True, CellMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultInlineFormatType->DefaultInputInlineFormatType, LineSpacing->{1.5, 1}, LimitsPositioningTokens->{}, FontFamily->"Times", FontSize->16], Cell[StyleData["SmallText", "Presentation"], CellMargins->{{24, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->16], Cell[StyleData["SmallText", "Printout"], CellMargins->{{14, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True], Cell[StyleData["SmallText", "TwoColumn"], CellMargins->{{14, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell["\<\ The cells in this section define styles used for input and output \ to the kernel. Be careful when modifying, renaming, or removing these \ styles, because the front end associates special meanings with these style \ names.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{45, Inherited}, {Inherited, Inherited}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, GroupPageBreakWithin->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", FontWeight->"Bold"], Cell[StyleData["Input", "Presentation"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{45, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Input", "Printout"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, FontSize->9], Cell[StyleData["Input", "TwoColumn"], CellFrame->{{3, 0}, {0, 0}}, CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, LineSpacing->{1.5, 0}, FormatType->StandardForm, FontFamily->"Courier", FontSize->14, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Output", "Presentation"], CellMargins->{{45, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Output", "Printout"], CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 0}, FontSize->10, FontColor->GrayLevel[0]], Cell[StyleData["Output", "TwoColumn"], CellMargins->{{30, Inherited}, {2, 2}}, PageBreakWithin->True, GroupPageBreakWithin->True, LineSpacing->{1, 0}, FontSize->12, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, StyleMenuListing->None, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Message", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Message", "Printout"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->9, FontColor->GrayLevel[0]], Cell[StyleData["Message", "TwoColumn"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->9, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Print", "Printout"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->9], Cell[StyleData["Print", "TwoColumn"], CellMargins->{{30, Inherited}, {2, 2}}, FontSize->16] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Info"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, StyleMenuListing->None], Cell[StyleData["Info", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Info", "Printout"], CellMargins->{{30, Inherited}, {Inherited, Inherited}}, FontSize->10], Cell[StyleData["Info", "TwoColumn"], CellMargins->{{30, Inherited}, {Inherited, Inherited}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{15, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, ImageMargins->{{35, Inherited}, {Inherited, 0}}, AnimationDisplayTime->0.2, StyleMenuListing->None, FontSize->14], Cell[StyleData["Graphics", "Presentation"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Graphics", "Printout"], CellMargins->{{30, Inherited}, {0, 0}}, CellFrameMargins->False, ImageSize->{Inherited, 150}, ImageMargins->{{45, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, FontSize->9], Cell[StyleData["Graphics", "TwoColumn"], CellMargins->{{20, Inherited}, {0, 0}}, CellFrameMargins->False, ImageSize->{Inherited, 150}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Times", FontSize->9, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["CellLabel", "Presentation"], FontSize->14], Cell[StyleData["CellLabel", "Printout"], FontColor->GrayLevel[1]], Cell[StyleData["CellLabel", "TwoColumn"], FontColor->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Special Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["PrefaceTitle"], CellFrame->{{1, 1}, {1, 5}}, ShowCellBracket->False, CellMargins->{{24, 24}, {0, 10}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, CellFrameMargins->{{15, Inherited}, {Inherited, Inherited}}, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, LineSpacing->{1.4, 1}, FontFamily->"Times", FontSize->24, FontWeight->"Bold", FontColor->GrayLevel[1], Background->RGBColor[0, 0.392187, 0]], Cell[StyleData["PrefaceTitle", "Presentation"], CellMargins->{{24, Inherited}, {60, Inherited}}, TextAlignment->Center, FontSize->38, FontColor->GrayLevel[1], Background->RGBColor[0.596078, 0.65098, 0.0196078]], Cell[StyleData["PrefaceTitle", "Printout"], CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["PrefaceTitle", "TwoColumn"], CellFrame->{{1, 1}, {0, 5}}, CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Special Body Text and Index", "Section"], Cell[CellGroupData[{ Cell[StyleData["Accident"], CellFrame->3, ShowCellBracket->False, CellMargins->{{24, 24}, {0, 10}}, CellFrameMargins->{{15, Inherited}, {Inherited, Inherited}}, TextAlignment->Center, LineSpacing->{1.4, 1}, FontFamily->"Times", FontSize->24, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], Cell[StyleData["Accident", "Presentation"], CellMargins->{{24, Inherited}, {60, Inherited}}, TextAlignment->Center, FontSize->36], Cell[StyleData["Accident", "Printout"], CellFrame->2, CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["Accident", "TwoColumn"], CellFrame->2, CellMargins->{{0, Inherited}, {0, Inherited}}, FontSize->16, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ContentsText"], CellMargins->{{50, 10}, {5, 5}}, FontFamily->"Times", FontSize->16], Cell[StyleData["ContentsText", "Presentation"]], Cell[StyleData["ContentsText", "Printout"], FontColor->GrayLevel[0], Background->None], Cell[StyleData["ContentsText", "TwoColumn"], FontColor->GrayLevel[0], Background->None] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special1"], CellDingbat->"\[EmptySquare]", ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, FontFamily->"Times", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.689998, 0.0899977, 0.119997]], Cell[StyleData["Special1", "Presentation"], FontSize->16], Cell[StyleData["Special1", "Printout"], FontSize->12, FontColor->GrayLevel[0]], Cell[StyleData["Special1", "TwoColumn"], FontSize->12, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special2"], CellMargins->{{6, 0}, {0, 0}}, CellGroupingRules->{"SectionGrouping", 40}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special2", "Presentation"], FontSize->12], Cell[StyleData["Special2", "Printout"], FontSize->10, FontColor->GrayLevel[0]], Cell[StyleData["Special2", "TwoColumn"], FontSize->10, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special3"], CellDingbat->"\[GraySquare]", ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Left, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special3", "Presentation"]], Cell[StyleData["Special3", "Printout"], FontColor->GrayLevel[0]], Cell[StyleData["Special3", "TwoColumn"], FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special4"], ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special4", "Presentation"], FontSize->12], Cell[StyleData["Special4", "Printout"], FontColor->GrayLevel[0]], Cell[StyleData["Special4", "TwoColumn"], FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Special5"], ShowClosedCellArea->True, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, TextAlignment->Center, FontFamily->"Courier", FontSize->10, FontColor->GrayLevel[0.333333]], Cell[StyleData["Special5", "Presentation"], FontSize->12], Cell[StyleData["Special5", "Printout"]], Cell[StyleData["Special5", "TwoColumn"]] }, Closed]], Cell[StyleData["IndexEntry"], ShowCellBracket->False, CellMargins->{{15, 5}, {0, 5}}, PageBreakBelow->False, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, ParagraphIndent->-40, StyleMenuListing->None, FontSize->16], Cell[StyleData["IndexSubEntry"], ShowCellBracket->False, CellMargins->{{45, 5}, {0, 0}}, DefaultFormatType->DefaultTextFormatType, DefaultInlineFormatType->DefaultInputInlineFormatType, ParagraphIndent->-40, StyleMenuListing->None, FontSize->16] }, Closed]], Cell[CellGroupData[{ Cell["Styles for License Agreement", "Section"], Cell[CellGroupData[{ Cell[StyleData["LicenseHeading"], ShowCellBracket->True, ShowGroupOpenCloseIcon->True, CellMargins->{{24, 24}, {-1, 2}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0.4, 0.300008, 0.6]], Cell[StyleData["LicenseHeading", "Presentation"], FontSize->12], Cell[StyleData["LicenseHeading", "Printout"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["LicenseHeading", "TwoColumn"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["LicenseText"], CellFrame->True, ShowCellBracket->False, CellMargins->{{24, 24}, {5, -1}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->12, Background->RGBColor[1, 0.537743, 0.509071]], Cell[StyleData["LicenseText", "Presentation"], FontSize->18], Cell[StyleData["LicenseText", "Printout"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[StyleData["LicenseText", "TwoColumn"], FontSize->10, FontColor->GrayLevel[0], Background->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Automatic Numbering", "Section"], Cell["\<\ The following styles are useful for numbered equations, figures, \ etc. They automatically give the cell a FrameLabel containing a reference to \ a particular counter, and also increment that counter.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["NumberedEquation"], CounterIncrements->"NumberedEquation"], Cell[StyleData["NumberedEquation", "Presentation"]], Cell[StyleData["NumberedEquation", "Printout"]], Cell[StyleData["NumberedEquation", "TwoColumn"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedFigure"], CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Figure ", CounterBox[ "NumberedFigure"]}]], None}}, CounterIncrements->"NumberedFigure", FormatTypeAutoConvert->False, FontFamily->"Times"], Cell[StyleData["NumberedFigure", "Presentation"], CellMargins->{{24, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["NumberedFigure", "Printout"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10], Cell[StyleData["NumberedFigure", "TwoColumn"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedTable"], CellMargins->{{6, Inherited}, {Inherited, Inherited}}, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Table ", CounterBox[ "NumberedTable"]}]], None}}, CounterIncrements->"NumberedTable", FormatTypeAutoConvert->False, FontFamily->"Times"], Cell[StyleData["NumberedTable", "Presentation"], CellMargins->{{24, Inherited}, {20, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["NumberedTable", "Printout"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10], Cell[StyleData["NumberedTable", "TwoColumn"], CellMargins->{{14, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontSize->10], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, StyleMenuListing->None, FontSize->9], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard \ ButtonFunctions, for use in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontSize->14, FontColor->RGBColor[0, 0.392187, 0], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["Hyperlink", "Printout"], FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False}], Cell[StyleData["Hyperlink", "TwoColumn"], FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["MEIHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.650004, 0.680003, 0.0800031], Background->RGBColor[0, 0.392187, 0], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Come visit us!"}], Cell[StyleData["MEIHyperlink", "Printout"], FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False}], Cell[StyleData["MEIHyperlink", "TwoColumn"], FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["BasicsHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.848096, 0.171878, 0.228321], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Go to Basics"}], Cell[StyleData["BasicsHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["BasicsHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["BasicsIndexHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", FontColor->RGBColor[0.848096, 0.171878, 0.228321], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2], FrontEnd`SelectionMove[ FrontEnd`SelectedNotebook[ ], Next, CellGroup], FrontEndToken[ "SelectionCloseAllGroups"], FrontEndToken[ "OpenCloseGroup"]}]&), Active->True, ButtonFrame->"None", ButtonNote->"Link into Basics"}], Cell[StyleData["BasicsIndexHyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}], Cell[StyleData["BasicsIndexHyperlink", "TwoColumn"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["TutorialsHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->None, FontWeight->"Bold", 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