// Numbas version: exam_results_page_options
{"name": "Indefinite integral", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Indefinite integral", "tags": ["Calculus", "Steps", "calculus", "constant of integration", "exponential function", "indefinite integration", "integrals", "integrating powers", "integration", "integration of exponential function", "integration of powers", "integration of trigonometric functions", "standard integrals", "steps", "trigonometric functions"], "advice": "\n  <p>Note that \\[\\begin{eqnarray*} &amp;\\int&amp; \\;x^n\\;dx&amp;=&amp;\\frac{x^{n+1}}{n+1}+C,\\;\\;n \\neq -1\\\\ &amp;\\int&amp; \\;\\sin(ax)\\;dx &amp;=&amp; -\\frac{1}{a}\\cos(ax)+C\\\\ &amp;\\int&amp; \\;e^{ax}\\;dx &amp;=&amp; \\frac{1}{a}e^{ax}+C\\\\ \\end{eqnarray*}\\]</p>\n  <p>Splitting the integral into three parts and using the above information we have: <br/>\\[\\begin{eqnarray*}\\simplify[std]{Int({b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3},x)}&amp;=&amp;\\simplify[std]{Int({b} * e ^ ({a}*x),x)+Int({b1} * Sin({a1}*x),x)+Int({a2} * x ^ {c3},x) }\\\\ &amp;=&amp;\\simplify[std]{({b}/{a}) * (e ^({a}*x)) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+C} \\end{eqnarray*}\\]</p>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n      <p>$\\simplify[std]{f(x) = {b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3}}$</p>\n        <p>$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]</p>\n        <p>Input all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.</p>\n        <p>Click on Show steps to get more information. You will not lose any marks by doing so.</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Input all numbers as integers or fractions and not decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [1.0, 2.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "({b}/{a}) * (e ^({a}*x)) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+C", "type": "jme"}], "steps": [{"prompt": "<p>Note that \\[\\begin{eqnarray*} &amp;\\int&amp; \\;x^n\\;dx&amp;=&amp;\\frac{x^{n+1}}{n+1}+C,\\;\\;n \\neq -1\\\\ &amp;\\int&amp; \\;\\sin(ax)\\;dx &amp;=&amp; -\\frac{1}{a}\\cos(ax)+C\\\\ &amp;\\int&amp; \\;e^{ax}\\;dx &amp;=&amp; \\frac{1}{a}e^{ax}+C\\\\ \\end{eqnarray*}\\]</p>", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "extensions": [], "statement": "\n  <p>Integrate the following function $f(x)$.</p>\n  <p>&nbsp;<br/>Input the constant of integration as $C$.</p>\n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "s1*random(2..5)", "name": "a"}, "b": {"definition": "s2*random(2..9)", "name": "b"}, "s3": {"definition": "random(1,-1)", "name": "s3"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "s5": {"definition": "random(1,-1)", "name": "s5"}, "s4": {"definition": "random(1,-1)", "name": "s4"}, "a1": {"definition": "random(2..5)", "name": "a1"}, "a2": {"definition": "s4*random(3..9)", "name": "a2"}, "b1": {"definition": "s3*random(2..9)", "name": "b1"}, "c3": {"definition": "s5*random(2..8)", "name": "c3"}}, "metadata": {"notes": "\n    \t\t<p><strong>2/08/2012:</strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p>Added description.</p>\n    \t\t<p>Corrected mistake in formula for integrating $\\sin(ax)$ in Steps and Advice.</p>\n    \t\t<p>Checked calculation. OK.</p>\n    \t\t<p>Added decimal point to forbidden strings along with message to user re input of numbers.</p>\n    \t\t<p>Message about Show steps included. Also another message about including the constant of integration.</p>\n    \t\t<p>Changed checking range from 0 to 1 to 1 to 2 as we can have negative powers of $x$.</p>\n    \t\t<p>Improved display of Steps by aligning integral signs.</p>\n    \t\t", "description": "<p>Find $\\displaystyle \\int ae ^ {bx}+ c\\sin(dx) + px ^ {q}\\;dx$.</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/"}]}