Perform the following integration, using the letter $C$ to represent any unknown constants.
", "advice": "integration by Susbtitution
", "extensions": [], "tags": [], "rulesets": {}, "metadata": {"description": "Integration by susbtitution, no hint given
", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"marks": "2", "variableReplacementStrategy": "originalfirst", "type": "jme", "vsetRangePoints": 5, "checkingType": "absdiff", "variableReplacements": [], "checkVariableNames": false, "showFeedbackIcon": true, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "malrules:\n [\n [\"ln({c}+x^2)/2\", \"Almost there! Did you forget to include the integration constant?\"],\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 $x$.\"],\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 $x$.\"],\n [\"ln(u)+C\", \"Double check the relationship between $du$ and $dx$. Did you sub in correctly here?\"],\n [\"ln(u)\", \"Double check the relationship between $du$ and $dx$. Did you sub in correctly here?\"],\n [\"ln({c}+x^2)+C\", \"Double check the relationship between $du$ and $dx$. Did you sub in correctly here?\"],\n [\"ln({c}+x^2)\", \"Double check the relationship between $du$ and $dx$. Did you sub in correctly here?\"],\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.\"],\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.\"],\n [\"1/(2u)\",\"You have not actually integrated anything. You simply substituted for ${c}+x^2$.\"],\n [\"1/(2u)+C\",\"You have not actually integrated anything. You simply substituted for ${c}+x^2$.\"]\n ]\n\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\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 )))