Find the following indefinite integral 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 [\"2/3*(1+e^x)^(3/2)\", \"Almost there! Did you forget to include the integration constant?\"],\n [\"e^x+2/3*e^(3x/2)\", \"$\\\\sqrt{1+e^x} \\\\neq \\\\sqrt{1}+\\\\sqrt{e^x}$. Hint: Try substitution.\"],\n [\"e^x+2/3*e^(3x/2)+C\", \"$\\\\sqrt{1+e^x} \\\\neq \\\\sqrt{1}+\\\\sqrt{e^x}$. Hint: Try substitution.\"],\n [\"2/3*(u)^(3/2)\", \"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 [\"2/3*(u)^(3/2)+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 [\"3/2*(1+e^x)^(3/2)\", \"You have multiplied by the new power rather than dividing by it.\"],\n [\"3/2*(1+e^x)^(3/2)+C\", \"You have multiplied by the new power rather than dividing by it.\"],\n [\"3/2*u^(3/2)+C\", \"You have multiplied by the new power rather than dividing by it.\"],\n [\"3/2*u^(3/2)\", \"You have multiplied by the new power rather than dividing by it.\"],\n [\"e^x*sqrt(x+e^x)+C\",\"You cannot simply integrate individual terms that are multiplied together. Hint: Try substitution.\"],\n [\"e^x*sqrt(x+e^x)\",\"You cannot simply integrate individual terms that are multiplied together. Hint: Try substitution.\"],\n [\"u^(1/2)+C\",\"You have not actually integrated anything. You simply substituted for $1+e^x$.\"],\n [\"u^(1/2)\",\"You have not actually integrated anything. You simply substituted for $1+e^x$.\"],\n [\"(1+e^x)^(1/2)+C\",\"You have not actually integrated anything. You simply substituted for $1+e^x$ and then subbed back in again.\"],\n [\"(1+e^x)^(1/2)\",\"You have not actually integrated anything. You simply substituted for $1+e^x$ and then subbed back in again.\"]\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 )))