// Numbas version: exam_results_page_options
{"name": "Integration: Indefinite integral", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s5*random(2..8)", "description": "", "name": "c3"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(3..9)", "description": "", "name": "a2"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s5"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(2..9)", "description": "", "name": "b"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s3"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(2..9)", "description": "", "name": "b1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..5)", "description": "", "name": "a"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "s4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s4"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "a1"}}, "ungrouped_variables": ["a", "b", "s3", "s2", "s1", "s5", "s4", "a1", "a2", "b1", "c3"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Integration: Indefinite integral", "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "({b}/{a}) * (e ^({a}*x)) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+C", "vsetrange": [1, 2], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input all numbers as integers or fractions and not decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t$\\simplify[std]{f(x) = {b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3}}$
\n\t\t\t $\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]
\n\t\t\t Input all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.
\n\t\t\t Click on Show steps to get more information. You will not lose any marks by doing so.
\n\t\t\t", "steps": [{"type": "information", "prompt": "Note that \\[\\begin{eqnarray*} &\\int& \\;x^n\\;dx&=&\\frac{x^{n+1}}{n+1}+C,\\;\\;n \\neq -1\\\\ &\\int& \\;\\sin(ax)\\;dx &=& -\\frac{1}{a}\\cos(ax)+C\\\\ &\\int& \\;e^{ax}\\;dx &=& \\frac{1}{a}e^{ax}+C\\\\ \\end{eqnarray*}\\]
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n\tIntegrate the following function $f(x)$.
\n\t
Input the constant of integration as $C$.
\n\t", "tags": ["calculus", "Calculus", "checked2015", "constant of integration", "exponential function", "indefinite integration", "integrating powers", "integration", "integration of exponential function", "integration of powers", "integration of trigonometric functions", "mas1601", "MAS1601", "standard integrals", "Steps", "steps", "trigonometric functions"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t2/08/2012:
\n\t\tAdded tags.
\n\t\tAdded description.
\n\t\tCorrected mistake in formula for integrating $\\sin(ax)$ in Steps and Advice.
\n\t\tChecked calculation. OK.
\n\t\tAdded decimal point to forbidden strings along with message to user re input of numbers.
\n\t\tMessage about Show steps included. Also another message about including the constant of integration.
\n\t\tChanged checking range from 0 to 1 to 1 to 2 as we can have negative powers of $x$.
\n\t\tImproved display of Steps by aligning integral signs.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\int ae ^ {bx}+ c\\sin(dx) + px ^ {q}\\;dx$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\tNote that \\[\\begin{eqnarray*} &\\int& \\;x^n\\;dx&=&\\frac{x^{n+1}}{n+1}+C,\\;\\;n \\neq -1\\\\ &\\int& \\;\\sin(ax)\\;dx &=& -\\frac{1}{a}\\cos(ax)+C\\\\ &\\int& \\;e^{ax}\\;dx &=& \\frac{1}{a}e^{ax}+C\\\\ \\end{eqnarray*}\\]
\n\t Splitting the integral into three parts and using the above information we have:
\\[\\begin{eqnarray*}\\simplify[std]{Int({b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3},x)}&=&\\simplify[std]{Int({b} * e ^ ({a}*x),x)+Int({b1} * Sin({a1}*x),x)+Int({a2} * x ^ {c3},x) }\\\\ &=&\\simplify[std]{({b}/{a}) * (e ^({a}*x)) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+C} \\end{eqnarray*}\\]
\n\t", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}