Integration by susbtitution, no hint given
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Find $\\displaystyle \\int \\frac{x}{\\var{c}+x^2} \\ dx$.
", "advice": "integration by Susbtitution
", "rulesets": {}, "extensions": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..8 except a)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["c", "b", "a"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "$\\displaystyle \\int \\frac{x}{\\var{c}+x^2} \\ dx = $ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "malrules:\n [\n [\"ln({c}+x^2)/2\", \"Almost there! Did you forget to include the integration constant?\",0.9],\n [\"1/2*ln(u)\", \"Don't forget to fill back in for $u$. You must give your answer in terms of the original variable i.e. in terms of $x$.\",0],\n [\"1/2*ln(u)+C\", \"Don't forget to fill back in for $u$. You must give your answer in terms of the original variable i.e. in terms of $x$.\",0],\n [\"ln(u)+C\", \"Double check the relationship between $du$ and $dx$. Did you sub in correctly here?\",0],\n [\"ln(u)\", \"Double check the relationship between $du$ and $dx$. Did you sub in correctly here?\",0],\n [\"ln({c}+x^2)+C\", \"Double check the relationship between $du$ and $dx$. Did you sub in correctly here?\",0],\n [\"ln({c}+x^2)\", \"Double check the relationship between $du$ and $dx$. Did you sub in correctly here?\",0],\n [\"1/(2({c}+x^2))\",\"You have not actually integrated anything. You simply substituted for ${c}+x^2$ and then subbed back in again.\",0],\n [\"1/(2({c}+x^2))+C\",\"You have not actually integrated anything. You simply substituted for ${c}+x^2$ and then subbed back in again.\",0],\n [\"1/(2u)\",\"You have not actually integrated anything. You simply substituted for ${c}+x^2$.\",0],\n [\"1/(2u)+C\",\"You have not actually integrated anything. You simply substituted for ${c}+x^2$.\",0]\n ]\n\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1],\"credit\":x[2]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))