// Numbas version: exam_results_page_options
{"name": "Definite Integration 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Definite Integration 2", "tags": ["Calculus", "calculus", "definite integration", "integration", "integration by parts", "integration by parts twice"], "advice": "\n  <p>&nbsp;</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>&nbsp;</p>\n  ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      <p>\\[I=\\int_0^{\\var{b1}}\\simplify[std]{e^({a}x)}\\;dx\\]</p>\n        <p>$I=\\;\\;$[[0]]</p>\n        <p>Input your answer to 3 decimal places.</p>\n      ", "gaps": [{"minvalue": "ans1-tol", "type": "numberentry", "maxvalue": "ans1+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>\\[I=\\int_0^{\\var{b2}}\\simplify[std]{1/({b}x+{m2})}\\;dx\\]</p>\n      <p>$I=\\;\\;$[[0]]</p>\n      <p>Input your answer to 3 decimal places.</p>\n      ", "gaps": [{"minvalue": "ans2-tol", "type": "numberentry", "maxvalue": "ans2+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>\\[I=\\int_0^{\\pi/2}\\simplify[std]{({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\\]</p>\n        <p>$I=\\;\\;$[[0]]</p>\n        <p>Input your answer to 3 decimal places.</p>\n      ", "gaps": [{"minvalue": "ans3-tol", "type": "numberentry", "maxvalue": "ans3+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "extensions": [], "statement": "<p>Evaluate the following definite integrals.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(-2..2#0.5 except 0)", "name": "a"}, "m2": {"definition": "random(1..9)", "name": "m2"}, "b": {"definition": "random(2..5)", "name": "b"}, "w": {"definition": "random(0,1)", "name": "w"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "ans2": {"definition": "precround(1/b*(ln(1+b*b2/m2)),3)", "name": "ans2"}, "ans3": {"definition": "precround(tans3,3)", "name": "ans3"}, "b2": {"definition": "random(1..20)", "name": "b2"}, "b1": {"definition": "random(-1..2#0.5 except 0)", "name": "b1"}, "tol": {"definition": 0.001, "name": "tol"}, "t": {"definition": "random(1,-1)", "name": "t"}, "m3": {"definition": "random(2..9)", "name": "m3"}, "ans1": {"definition": "precround(tans1,3)", "name": "ans1"}, "c1": {"definition": "t*random(1..9)", "name": "c1"}, "tans1": {"definition": "(1/a)*(e^(a*b1)-1)", "name": "tans1"}, "tol1": {"definition": 0.0001, "name": "tol1"}, "tans3": {"definition": "1/m3*((1-w)*sin(m3*pi/2)-w*(cos(m3*pi/2)-1))", "name": "tans3"}, "d1": {"definition": "random(-9..9)", "name": "d1"}}, "metadata": {"notes": "\n    \t\t                \t\t<p><strong>3/07/1012:</strong></p>\n    \t\t                \t\t        <p>Added tags.</p>\n    \t\t                \t\t        <p>Checked calculations.</p>\n    \t\t                \t\t        <p>Left tolerances in, as easy to make minor errors in calculations.</p>\n    \t\t                \t\t        <p>Improved display in Advice.</p>\n    \t\t                \t\t        <p>Some superscripts e.g. the form a^\\var{b} in latex have to be written as a^{\\var{b}} as not displayed properly (if b has a second digit it slips down). Corrected.</p>\n    \t\t                \t\t        <p><strong>20/07/2012:</strong></p>\n    \t\t                \t\t        <p>Set new tolerace variables, tol=0.01, tol1=0.0001.</p>\n    \t\t                \t\t        <p>Can have expressions in Advice of the form $1\\times E$ where E is an expression. This can be remedied by rewriting - but later as not crucial.</p>\n    \t\t                \t\t        <p>Added description.</p>\n    \t\t                \t\t        <p></p>\n    \t\t                \t\t        <p><strong>25/07/2012:</strong></p>\n    \t\t                \t\t        <p>&nbsp;</p>\n    \t\t                \t\t        <p>Added tags.</p>\n    \t\t                \t\t        <p>A lot of work in this question - Perhaps it would be more managable broken down into two separate questions?</p>\n    \t\t                \t\t        <p>&nbsp;</p>\n    \t\t                \t\t        <p>Question appears to be working correctly.</p>\n    \t\t                \t\t        <p>&nbsp;</p>\n    \t\t                \t\t        <p>&nbsp;</p>\n    \t\t                \n    \t\t", "description": "<p>Evaluate $\\int_0^{\\,m}e^{ax}\\;dx$, $\\int_0^{p}\\frac{1}{bx+d}\\;dx,\\;\\int_0^{\\pi/2} \\sin(qx) \\;dx$.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}