(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 25345, 873] NotebookOptionsPosition[ 22513, 783] NotebookOutlinePosition[ 22927, 801] CellTagsIndexPosition[ 22884, 798] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Lesson 6 Differentiation and Integration", "Section", CellChangeTimes->{{3.461019705701769*^9, 3.461019711482246*^9}}], Cell["\<\ In this lesson you will learn about the built-in functions related to \ differentiation and integration.\ \>", "Text"], Cell[CellGroupData[{ Cell["Differentiation", "Subsection"], Cell[TextData[{ "The built-in function for differentiation is abbreviated ", StyleBox["D", "Input", FontWeight->"Bold"], "." }], "Text", CellChangeTimes->{3.461074352611628*^9}], Cell[BoxData[ RowBox[{"?", "D"}]], "Input"], Cell["Here are some examples:", "Text", CellChangeTimes->{{3.46101976874737*^9, 3.461019833725844*^9}, { 3.4610750752211514`*^9, 3.461075078116417*^9}}], Cell[BoxData[ RowBox[{"D", "[", RowBox[{ RowBox[{"x", "^", "2"}], ",", "x"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"D", "[", RowBox[{ RowBox[{"x", "^", "n"}], ",", "x"}], "]"}]], "Input", CellChangeTimes->{{3.4610743847150917`*^9, 3.461074389656864*^9}}], Cell[BoxData[ RowBox[{"D", "[", RowBox[{ RowBox[{"a", " ", RowBox[{"Sin", "[", RowBox[{"b", " ", "x"}], "]"}]}], ",", "x"}], "]"}]], "Input", CellChangeTimes->{{3.461074359624703*^9, 3.4610743734096737`*^9}}], Cell[TextData[{ "In each case we are differentiating with respect to ", StyleBox["x", "Input"], ". Notice that all other undefined variables are assumed to be constants and \ have zero derivative with respect to ", StyleBox["x", "Input"], "." }], "Text", CellChangeTimes->{{3.461074954663197*^9, 3.461075068133931*^9}}], Cell[TextData[{ "If there is more than one variable, then ", StyleBox["D", "Input"], " gives partial derivatives." }], "Text", CellChangeTimes->{{3.461075094658215*^9, 3.461075151804015*^9}, { 3.4610828100294313`*^9, 3.461082832622714*^9}}], Cell[BoxData[{ RowBox[{"f", "=", RowBox[{"Sin", "[", RowBox[{"a", " ", "x", " ", RowBox[{"E", "^", "y"}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{"D", "[", RowBox[{"f", ",", "x"}], "]"}], "\[IndentingNewLine]", RowBox[{"D", "[", RowBox[{"f", ",", "y"}], "]"}]}], "Input", CellChangeTimes->{ 3.461074417102953*^9, {3.461075162302569*^9, 3.461075201251198*^9}, { 3.461075235358453*^9, 3.461075235639998*^9}}], Cell[TextData[{ "In calculating ", StyleBox["D[f, x]", "Input"], ", ", StyleBox["Mathematica", FontSlant->"Italic"], " treats both ", StyleBox["a", "Input"], " and ", StyleBox["y", "Input"], " as constants." }], "Text", CellChangeTimes->{{3.4610752507492847`*^9, 3.461075295828532*^9}}], Cell[TextData[{ "To get higher derivatives, we need to specify the order. If we want to \ compute the third derivative of ", StyleBox["x^(1/3)", "Input"], ", we use" }], "Text"], Cell[BoxData[ RowBox[{"D", "[", RowBox[{ RowBox[{"x", "^", RowBox[{"(", RowBox[{"1", "/", "3"}], ")"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "3"}], "}"}]}], "]"}]], "Input"], Cell[TextData[{ "Finally, for functions of several variables, we can compute mixed partial \ derivatives. If we want to compute the partial derivative once with respect \ to ", StyleBox["x", "Input"], " and twice with respect to ", StyleBox["y", "Input"], ", we need to list each variable with its respective order as a separate \ list. If the order is ", StyleBox["1", "Input"], ", we can leave out the order (and the list)" }], "Text", CellChangeTimes->{{3.461074474610574*^9, 3.461074476317465*^9}}], Cell[BoxData[ RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], " ", RowBox[{"y", "^", "3"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", "2"}], "}"}]}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"D", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], " ", RowBox[{"y", "^", "3"}]}], ",", "x", ",", RowBox[{"{", RowBox[{"y", ",", "2"}], "}"}]}], "]"}]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Practice", "Subsection", CellFrame->{{0, 0}, {0, 2}}, CellChangeTimes->{{3.461075991370789*^9, 3.4610759933204603`*^9}}], Cell[TextData[{ "1) Find the critical numbers of", StyleBox[" ", "Input"], StyleBox[Cell[BoxData[ FormBox[ RowBox[{"f", "=", RowBox[{ SuperscriptBox["x", "2"], SuperscriptBox["\[ExponentialE]", "x"]}]}], TraditionalForm]], "Input"], "Input"], "." }], "Text", CellChangeTimes->{{3.461076009228404*^9, 3.4610760576025*^9}, 3.4610761218915052`*^9}], Cell[TextData[{ "2) Find the inflection points of", StyleBox[" ", "Input"], StyleBox[Cell[BoxData[ FormBox[ RowBox[{"f", "=", RowBox[{ SuperscriptBox["x", "2"], SuperscriptBox["\[ExponentialE]", "x"]}]}], TraditionalForm]], "Input"], "Input"], ". Can you do this in one line?" }], "Text", CellChangeTimes->{{3.461076009228404*^9, 3.4610760576025*^9}, 3.4610761218915052`*^9, {3.4610762047787113`*^9, 3.461076280435783*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Indefinite Integration", "Subsection", CellFrame->{{0, 0}, {0, 2}}, CellChangeTimes->{{3.4610745756297483`*^9, 3.461074578523662*^9}}], Cell[TextData[{ "You will remember from calculus that differentiation is easy and \ integration is hard. So it is good that ", StyleBox["Mathematica", FontSlant->"Italic"], " knows all of the tricks that you have learned about integration, and a lot \ more. The function that does the work is called ", StyleBox["Integrate", "Input"], "." }], "Text", CellChangeTimes->{{3.416498846738912*^9, 3.416498848687447*^9}, 3.447973287889086*^9, 3.447973524883885*^9, {3.461074647072446*^9, 3.461074870844041*^9}, {3.461075343182088*^9, 3.461075350565763*^9}}], Cell[BoxData[ RowBox[{"?", "Integrate"}]], "Input"], Cell[TextData[{ "Let's compute the following indefinite integral:\n\t\t\t\t", Cell[BoxData[ FormBox[ RowBox[{"\[Integral]", RowBox[{ RowBox[{ SuperscriptBox["x", "4"], " ", RowBox[{"cos", "(", "x", ")"}]}], RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]], FontWeight->"Bold"], StyleBox[".", FontSize->16] }], "Text", CellChangeTimes->{{3.416498846738912*^9, 3.416498848687447*^9}, 3.447973287889086*^9, 3.447973524883885*^9, {3.461074647072446*^9, 3.4610748321240807`*^9}, {3.461074884973571*^9, 3.46107488918766*^9}, { 3.4610753628374557`*^9, 3.461075373512869*^9}, {3.461082880536075*^9, 3.461082884829791*^9}}], Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "4"}], " ", RowBox[{"Cos", "[", "x", "]"}]}], ",", "x"}], "]"}]], "Input"], Cell[TextData[{ "Notice that we need to specify the integrand and the variable of \ integration, and that ", StyleBox["Mathematica", FontSlant->"Italic"], " does not include an arbitrary constant of integration (", StyleBox["+C", "Input"], ")." }], "Text", CellChangeTimes->{{3.4610753761628428`*^9, 3.4610754157082233`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Practice", "Subsection", CellFrame->{{0, 0}, {0, 2}}, CellChangeTimes->{{3.4610749354171762`*^9, 3.461074941582384*^9}, { 3.4610754496285267`*^9, 3.461075467461878*^9}}], Cell[TextData[{ "1) Calculate the indefinite integral of ", StyleBox[Cell[BoxData[ FormBox[ SuperscriptBox["x", "n"], TraditionalForm]], "Input"], "Input"], " with respect to ", StyleBox["x", "Input"], ". Now differentiate the result. Do you get ", StyleBox[Cell[BoxData[ FormBox[ SuperscriptBox["x", "n"], TraditionalForm]], "Input"], "Input"], "back?" }], "Text", CellChangeTimes->{{3.461075470987918*^9, 3.4610754993645077`*^9}, { 3.4610755690677767`*^9, 3.4610755738844423`*^9}, {3.4610756436000032`*^9, 3.4610757338755836`*^9}}], Cell[TextData[{ "2) Calculate the indefinite integral of ", StyleBox[Cell[BoxData[ FormBox[ SuperscriptBox["x", "n"], TraditionalForm]], "Input"], "Input"], " with respect to ", StyleBox["n", "Input"], ". Now differentiate the result. Do you get ", StyleBox[Cell[BoxData[ FormBox[ SuperscriptBox["x", "n"], TraditionalForm]], "Input"], "Input"], "back?" }], "Text", CellChangeTimes->{{3.461075470987918*^9, 3.4610754993645077`*^9}, { 3.4610755690677767`*^9, 3.4610755738844423`*^9}, {3.4610756436000032`*^9, 3.4610757338755836`*^9}, {3.4610757749871473`*^9, 3.461075778232018*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Definite Integrals", "Subsection", CellFrame->{{0, 0}, {0, 2}}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can also compute definite integrals. In addition to the integrand and the \ variable of integration, the limits of integration have to be specified. \ Either or both of the limits can be infinite, entered either as ", StyleBox["\[Infinity]", "Input"], " from the palette or spelled out as ", StyleBox["Infinity", "Input", FontWeight->"Bold"], ".", StyleBox[" ", "MR"], StyleBox["Let's calculate the following integrals.", "MR", FontFamily->"Times New Roman"], "\n\n\t\t", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "10"], RowBox[{"x", RowBox[{"\[DifferentialD]", "x"}]}]}]]], ", ", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", "x"}]], RowBox[{"\[DifferentialD]", "x"}]}]}]]], ", and ", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "t"], RowBox[{ SuperscriptBox["x", "2"], RowBox[{"\[DifferentialD]", "x"}]}]}]]], "." }], "Text", CellChangeTimes->{{3.461076410330217*^9, 3.46107645324867*^9}}], Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{"x", ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "10"}], "}"}]}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{"E", "^", RowBox[{"(", RowBox[{"-", "x"}], ")"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "Infinity"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.461076468438855*^9, 3.461076469960041*^9}}], Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{"x", "^", "2"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "t"}], "}"}]}], "]"}]], "Input"], Cell[TextData[{ "Here's one that ", StyleBox["Mathematica", FontSlant->"Italic"], " has to think about a bit.\n\n", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[{ SuperscriptBox["e", RowBox[{"-", "x"}]], SuperscriptBox["x", RowBox[{"3", "/", "2"}]], RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.461083711764236*^9, 3.4610837943739967`*^9}}], Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ RowBox[{"E", "^", RowBox[{"(", RowBox[{"-", "x"}], ")"}]}], " ", RowBox[{"x", "^", RowBox[{"(", RowBox[{"3", "/", "2"}], ")"}]}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "Infinity"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.461082953777464*^9, 3.4610830512286253`*^9}, { 3.461083092976994*^9, 3.4610830997148247`*^9}}], Cell["\<\ Here are two integrals whose values are expressed in terms of functions you \ may not know.\ \>", "Text", CellChangeTimes->{{3.461083807728436*^9, 3.461083923329801*^9}}], Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{"E", "^", RowBox[{"(", RowBox[{"-", RowBox[{"x", "^", "2"}]}], ")"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.461083113421083*^9, 3.4610831342106524`*^9}}], Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{"Sin", "[", RowBox[{"x", "^", "2"}], "]"}], ",", "x"}], "]"}]], "Input"], Cell[TextData[{ "But even ", StyleBox["Mathematica", FontSlant->"Italic"], " can't find an expression for every integral:" }], "Text", CellChangeTimes->{{3.461082918584421*^9, 3.461082942280491*^9}, { 3.461083147784213*^9, 3.461083150894496*^9}, {3.461083983594475*^9, 3.461083985816099*^9}}], Cell[BoxData[ RowBox[{"Integrate", "[", " ", RowBox[{ RowBox[{ RowBox[{"Sin", "[", RowBox[{ RowBox[{"x", "^", "3"}], "+", "1"}], "]"}], RowBox[{"Exp", "[", RowBox[{"x", "+", "1"}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.461083159405855*^9, 3.461083243605547*^9}, { 3.461083274846348*^9, 3.461083406387622*^9}, {3.461083949872691*^9, 3.46108395060452*^9}}], Cell[TextData[{ "For such cases, ", StyleBox["Mathematica", FontSlant->"Italic"], " can provide an approximation using the ", StyleBox["NIntegrate", "Input"], " function. The syntax for this function is identical to the one for ", StyleBox["Integrate", "Input", FontWeight->"Bold"], "." }], "Text", CellChangeTimes->{{3.461083439218412*^9, 3.461083506470693*^9}}], Cell[BoxData[ RowBox[{"NIntegrate", "[", " ", RowBox[{ RowBox[{ RowBox[{"Sin", "[", RowBox[{ RowBox[{"x", "^", "3"}], "+", "1"}], "]"}], RowBox[{"Exp", "[", RowBox[{"x", "+", "1"}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.461083159405855*^9, 3.461083243605547*^9}, { 3.461083274846348*^9, 3.461083406387622*^9}, {3.461083949872691*^9, 3.4610839696655273`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Practice", "Subsection", CellFrame->{{0, 0}, {0, 2}}], Cell[TextData[{ "1) Let", StyleBox[" ", "Output"], StyleBox[Cell[BoxData[ RowBox[{"f", "=", FractionBox[ SuperscriptBox["x", "3"], RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["x", "2"], "+", "2"}], ")"}], RowBox[{"(", RowBox[{"x", "-", "3"}], ")"}]}]]}]], "Output", CellChangeTimes->{{3.4610758512545*^9, 3.461075895345809*^9}}], "Output"], ". Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to integrate this function and then differentiate the result. Do you get", StyleBox[" f", "Input"], " back?" }], "Text", CellChangeTimes->{{3.4479730978731937`*^9, 3.447973100351673*^9}, { 3.4610758236248493`*^9, 3.461075824459208*^9}, 3.461075929992999*^9, { 3.46374780733801*^9, 3.4637478101522207`*^9}, 3.4637478415527267`*^9}], Cell[TextData[{ "2) Let ", StyleBox[Cell[BoxData[ RowBox[{"f", "=", FractionBox[ SuperscriptBox["x", "3"], RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["x", "2"], "+", "2"}], ")"}], RowBox[{"(", RowBox[{"x", "-", "3"}], ")"}]}]]}]], "Output", CellChangeTimes->{{3.4610758512545*^9, 3.461075895345809*^9}}], "Output"], ". Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to diferentiate this function and then integrate the result. Do you get ", StyleBox["f", "Input"], " back?" }], "Text", CellChangeTimes->{{3.461075829723483*^9, 3.461075830138942*^9}, 3.461075972458139*^9, {3.463747814456354*^9, 3.463747838344651*^9}}], Cell[TextData[{ "3) Evaluate the polynomial ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ RowBox[{"f", "(", "x", ")"}], "Input"], StyleBox["=", "Input"], RowBox[{ StyleBox[ RowBox[{ RowBox[{"2", SuperscriptBox["x", "3"]}], "+", RowBox[{"9", SuperscriptBox["x", "2"]}], "-", RowBox[{"30", "x"}], "+", "4"}], "Input"], " ", "at", " ", "its", " ", "critical", " ", RowBox[{"numbers", "."}]}]}], TraditionalForm]]] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Homework", "Subsection", CellFrame->{{0, 0}, {0, 2}}, CellChangeTimes->{{3.46108414239854*^9, 3.46108414533778*^9}}], Cell[TextData[{ "1) ", "Find the following derivatives.\n (a) the second derivative of ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ SuperscriptBox["sin", "2"], "(", "x", ")"}], "Input"], TraditionalForm]], "Input"], "." }], "Text", CellChangeTimes->{{3.461084570791834*^9, 3.461084578883746*^9}, 3.461085425674625*^9}], Cell[BoxData[ RowBox[{ RowBox[{"(", "b", ")"}], " ", StyleBox[Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox["d", "2"], RowBox[{ StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]], StyleBox[" ", FontSlant->"Italic"], "dy"}]], TraditionalForm]], "Input"], "Input"], StyleBox[Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"2", "x"}]], RowBox[{"sin", "(", "xy", ")"}]}], TraditionalForm]], "Input"], "Input"]}]], "Text", CellChangeTimes->{{3.4610854519138002`*^9, 3.461085455000905*^9}}, FontColor->GrayLevel[0]], Cell[TextData[{ "2) Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to compute the ", StyleBox["exact", FontWeight->"Bold"], " values for the integral of the following expressions. Be aware that ", StyleBox["Mathematica", FontSlant->"Italic"], " can find closed form solutions for many, not all integrals.\n(a) ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{" ", RowBox[{"1", "/", SqrtBox[ RowBox[{ RowBox[{"2", "a", " ", "x"}], "+", SuperscriptBox["x", "2"]}]]}]}], "Input"], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.448195852226054*^9, 3.44819585222609*^9}, { 3.448195893203958*^9, 3.448195911037004*^9}, {3.448195966891161*^9, 3.448195982599103*^9}, {3.46108521681492*^9, 3.461085217411747*^9}}, FontWeight->"Plain"], Cell[TextData[{ "(b)", Cell[BoxData[ FormBox[ RowBox[{" ", StyleBox[ RowBox[{"1", "/", SqrtBox[ RowBox[{ RowBox[{"2", "a", " ", "x"}], "-", SuperscriptBox["x", "2"]}]]}], "Input"]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.448195852226054*^9, 3.44819585222609*^9}, { 3.448195893203958*^9, 3.448195911037004*^9}, {3.448195986602634*^9, 3.448195989005582*^9}, {3.448196036341547*^9, 3.448196063067308*^9}}, FontWeight->"Plain"], Cell[TextData[{ "(c) ", Cell[BoxData[ FormBox[ RowBox[{"1", "/", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["sin", "2"], "(", RowBox[{"a", " ", "x"}], ")"}], RowBox[{ SuperscriptBox["cos", "2"], "(", RowBox[{"a", " ", "x"}], ")"}]}], ")"}]}], TraditionalForm]], "Input"] }], "Text", CellChangeTimes->{{3.418228089414297*^9, 3.418228098509897*^9}, 3.461084965833276*^9}, FontWeight->"Plain"], Cell[TextData[{ "3) ", StyleBox["Obtain ", FontVariations->{"CompatibilityType"->0}], StyleBox["approximate numerical", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], StyleBox[" values for the integrals below. ", FontVariations->{"CompatibilityType"->0}], "Be careful to use the proper ", StyleBox["Mathematica", FontSlant->"Italic"], " functions for ln and ", Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox["tan", RowBox[{"-", "1"}]]}], TraditionalForm]]], "(look them up!)." }], "Text", CellChangeTimes->{{3.461084702208061*^9, 3.4610847138696012`*^9}, { 3.4610848374603148`*^9, 3.461084845260159*^9}, 3.46108522919835*^9}], Cell[TextData[{ "(a) ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[{ RowBox[{"sin", "(", "x", ")"}], " ", SuperscriptBox["e", RowBox[{"-", SuperscriptBox["x", "2"]}]], RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]], "Input"] }], "Text"], Cell[TextData[{ "(b) ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "1", RowBox[{"\[Pi]", " "}]], RowBox[{ RowBox[{"ln", "(", "x", ")"}], " ", RowBox[{ SuperscriptBox["tan", RowBox[{"-", "1"}]], "(", RowBox[{"2", SqrtBox["x"]}], ")"}], RowBox[{"\[DifferentialD]", "x"}]}]}], "Input"], TraditionalForm]]] }], "Text"], Cell[TextData[{ "4) Integrate the quantity ", StyleBox[Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"1", "/", RowBox[{"(", RowBox[{ SuperscriptBox["x", "3"], "+", "1"}], ")"}]}], "Input"], TraditionalForm]], "Input"], "Input"], " with respect to ", Cell[BoxData[ FormBox[ StyleBox["x", "Input"], TraditionalForm]]], ". Use the ", StyleBox["D", "Input", FontWeight->"Bold"], " function to differentiate the result you obtained. Show that the result \ you obtain when differentiating is equal to ", StyleBox[Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"1", "/", RowBox[{"(", RowBox[{ SuperscriptBox["x", "3"], "+", "1"}], ")"}]}], "Input"], TraditionalForm]], "Input"], "Input"], " by using ", StyleBox["Simplify", "Input", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], "or a similar function to modify the result of the differentiation." }], "Text", CellChangeTimes->{{3.448195852226054*^9, 3.44819585222609*^9}, { 3.448195893203958*^9, 3.448195911037004*^9}, {3.448195986602634*^9, 3.448195989005582*^9}, {3.448196124335108*^9, 3.448196124818327*^9}, { 3.448196497266223*^9, 3.448196500960291*^9}, {3.461085133062455*^9, 3.461085177827814*^9}, 3.461085232106866*^9, 3.461085490843808*^9, 3.46452519567211*^9}, FontWeight->"Plain"], Cell[TextData[{ "5) Create a table containing the values of", StyleBox[" ", "Input"], StyleBox[Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[{ SuperscriptBox["x", "n"], SuperscriptBox["e", RowBox[{"-", "x"}]], RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]], "Input"], "Input"], " for ", StyleBox["n=1,2,...,10", "Input"], ". 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