// Numbas version: exam_results_page_options
{"name": "cormac's copy of Julie's copy of Definite integration 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": ["Calculus", "calculus", "definite integration", "integration", "integration by parts", "integration by parts twice", "rebelmaths"], "name": "cormac's copy of Julie's copy of Definite integration 1", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "advice": "\n  <p>a)<br/>\\[I=\\int_1^\\var{b1}\\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\\;dx\\] <br/>Expand the parentheses to obtain:</p>\n<p>\\[\\begin{eqnarray*}I &amp;=&amp; \\int_1^\\var{b1} \\simplify[std]{{a1 ^ 2} * x ^ 4 + {2 * a1 * c1} * x ^ 3+ {c1 ^ 2+2*a1*d1} * x ^ 2 + {2 * c1 * d1} * x+ {d1 ^ 2} }\\;dx\\\\ &amp;=&amp;\\left[\\simplify[std]{{a1 ^ 2}/5 * x ^ 5 + {2 * a1 * c1}/4 * x ^ 4+ {c1 ^ 2+2*a1*d1}/3 * x ^ 3 + {2 * c1 * d1}/2 * x^2+ {d1 ^ 2}x }\\right]_1^\\var{b1}\\\\ &amp;=&amp;\\var{tans1}\\\\ \\\\&amp;=&amp;\\var{ans1}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]</p>\n<p>b)<br/>\\[\\begin{eqnarray*}I&amp;=&amp;\\int_0^{\\var{b2}}\\simplify[std]{1/(x+{m2})}\\;dx\\\\ &amp;=&amp;\\left[\\ln(x+\\var{m2})\\right]_0^{\\var{b2}}\\\\ &amp;=&amp; \\ln(\\var{b2+m2})-\\ln(\\var{m2})\\\\ &amp;=&amp;\\ln\\left(\\frac{\\var{b2+m2}}{\\var{m2}}\\right)\\\\ &amp;=&amp;\\var{ans2}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]</p>\n<p>c)<br/>\\[I=\\int_0^\\pi\\simplify[std]{x * ({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\\] <br/>We use integration by parts.</p>\n<p>Recall that:<br/>\\[\\int u\\frac{dv}{dx}\\;dx=uv-\\int \\frac{du}{dx}\\;v\\;dx\\]<br/>Here we set $u=x$ and $\\displaystyle \\frac{dv}{dx}=\\simplify[std]{ {w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x)}$</p>\n<p>Hence \\[v=\\simplify[std]{({-w}/ {m3}) * Cos({m3} * x) + {1 -w} * (({1-w}/ {m3}) * Sin({m3} * x))}\\]</p>\n<p>So \\[\\begin{eqnarray*} I&amp;=&amp;\\left[\\simplify[std]{{-w}*((x / {m3}) * Cos({m3} * x)) + {1 -w} * ((x / {m3}) * Sin({m3} * x))}\\right]_0^\\pi -\\int_0^\\pi\\simplify[std]{ ({ -w} / {m3} )* Cos({m3} * x) + {1 -w} * (1 / {m3} * Sin({m3} * x))}\\;dx\\\\ &amp;=&amp;\\simplify[std]{({-w*cos(m3*pi)})*({pi}/{m3})}-\\left[\\simplify[std]{{ -w} * (1 / {m3 ^ 2})* Sin({m3} * x) -({1 -w} * (1 / {m3 ^ 2}) * Cos({m3} * x))}\\right]_0^\\pi\\\\ &amp;=&amp; \\var{ans3}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]<br/>d)</p>\n<p>\\[I=\\int_0^{\\var{b4}}\\simplify[std]{x ^ {m4} * Exp({n4} * x)}\\;dx\\]</p>\n<p>Use integration by parts twice with $u=x^2$ and $\\displaystyle \\frac{dv}{dx}=\\simplify[std]{e^({n4}x)}\\Rightarrow v = \\simplify[std]{1/{n4}e^({n4}x)}$<br/>\\[\\begin{eqnarray*} I&amp;=&amp;\\left[\\simplify[std]{x^2/{n4}Exp({n4} * x)}\\right]_0^{\\var{b4}}+\\simplify[std]{2/{abs(n4)}DefInt(x*Exp({n4} * x),x,0,{b4})}\\\\ &amp;=&amp;\\simplify[std]{{b4^2}/{n4}*e^{p}-2/{n4}}\\left(\\left[\\simplify[std]{x/{n4}*e^({n4}*x)}\\right]_0^{\\var{b4}}+\\simplify[std]{1/{abs(n4)}DefInt(e^({n4}x),x,0,{b4})}\\right)\\\\ &amp;=&amp;\\simplify[std]{{b4^2}/{n4}*e^{p}-2/{n4}}\\left(\\simplify[std]{{b4}/{n4}*e^{p}-1/{n4}}\\left[\\simplify[std]{1/{n4}*e^({n4}*x)}\\right]_0^{\\var{b4}}\\right)\\\\ &amp;=&amp;\\simplify[std]{({b4 ^ 2} / {n4}) * Exp({p}) -(({2 * b4} / {n4 ^ 2}) * Exp({p})) + (2 / {n4 ^ 3}) * (Exp({p}) -1)}\\\\ &amp;=&amp;\\var{ans4}\\mbox{ to 4 decimal places} \\end{eqnarray*} \\]</p>\n  \n", "parts": [{"showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"showCorrectAnswer": true, "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "ans1+tol", "allowFractions": false, "minValue": "ans1-tol", "type": "numberentry", "scripts": {}, "marks": 4, "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false}], "marks": 0, "prompt": "\n  <p>\\[I=\\int_1^{\\var{b1}}\\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\\;dx\\]</p>\n<p>$I=\\;\\;$[[0]]</p>\n<p>Input your answer to 2 decimal places.</p>\n  \n", "type": "gapfill", "scripts": {}, "variableReplacementStrategy": "originalfirst"}], "variable_groups": [], "question_groups": [{"pickQuestions": 0, "name": "", "pickingStrategy": "all-ordered", "questions": []}], "variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": [], "showQuestionGroupNames": false, "ungrouped_variables": ["ans1", "ans2", "ans3", "ans4", "b4", "b1", "b2", "d1", "s2", "s7", "s6", "m4", "m3", "m2", "tol", "a1", "tans4", "c1", "tans1", "tans3", "tol1", "p", "t", "w", "n4"], "variables": {"d1": {"definition": "random(-9..9)", "group": "Ungrouped variables", "name": "d1", "description": "", "templateType": "anything"}, "b1": {"definition": "random(2..6)", "group": "Ungrouped variables", "name": "b1", "description": "", "templateType": "anything"}, "tol": {"definition": "0.01", "group": 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"ans2", "description": "", "templateType": "anything"}, "p": {"definition": "n4*b4", "group": "Ungrouped variables", "name": "p", "description": "", "templateType": "anything"}, "m4": {"definition": "2", "group": "Ungrouped variables", "name": "m4", "description": "", "templateType": "anything"}, "c1": {"definition": "t*random(1..9)", "group": "Ungrouped variables", "name": "c1", "description": "", "templateType": "anything"}, "a1": {"definition": "random(1..7)", "group": "Ungrouped variables", "name": "a1", "description": "", "templateType": "anything"}, "tans3": {"definition": "if(w=0,((-1)^(m3)-1)/m3^2,-pi*(-1)^(m3)/m3)", "group": "Ungrouped variables", "name": "tans3", "description": "", "templateType": "anything"}, "tol1": {"definition": "0.0001", "group": "Ungrouped variables", "name": "tol1", "description": "", "templateType": "anything"}, "s7": {"definition": "1", "group": "Ungrouped variables", "name": "s7", "description": "", "templateType": "anything"}, "ans3": {"definition": "precround(tans3,2)", "group": "Ungrouped variables", "name": "ans3", "description": "", "templateType": "anything"}}, "metadata": {"description": "<p>Evaluate $\\int_1^{\\,m}(ax ^ 2 + b x + c)^2\\;dx$, $\\int_0^{p}\\frac{1}{x+d}\\;dx,\\;\\int_0^\\pi x \\sin(qx) \\;dx$, $\\int_0^{r}x ^ {2}e^{t x}\\;dx$</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "statement": "<p>Evaluate the following definite integral.</p>", "type": "question", "preamble": {"js": "", "css": ""}, "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}]}]}], "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}]}