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*}\\]
", "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst"}], "prompt": "\n$\\simplify[std]{f(x) = {b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3}}$
\n$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]
\nEnter all numbers as integers or fractions and not as decimals.
\n ", "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"type": "jme", "checkingaccuracy": 0.001, "vsetrange": [0, 1], "notallowed": {"strings": ["."], "partialCredit": 0, "showStrings": false, "message": "Enter all numbers as integers or fractions and not as decimals.
"}, "expectedvariablenames": [], "scripts": {}, "marks": 3, "showpreview": true, "checkvariablenames": false, "answer": "({b}/{a}) * e ^({a}*x) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+C", "showCorrectAnswer": true, "vsetrangepoints": 5, "checkingtype": "absdiff", "answersimplification": "std", "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}], "stepsPenalty": 0, "marks": 0, "variableReplacementStrategy": "originalfirst"}], "question_groups": [{"pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [], "name": ""}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "showQuestionGroupNames": false, "ungrouped_variables": ["a", "b", "s3", "s2", "s1", "s5", "s4", "a1", "a2", "b1", "c3"], "variable_groups": [], "advice": "\nSplitting the integral into three parts and using the information in Steps we have:
\n\\[\\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 ", "variables": {"a1": {"definition": "random(2..5)", "group": "Ungrouped variables", "name": "a1", "description": "", "templateType": "anything"}, "b": {"definition": "s2*random(2..9)", "group": "Ungrouped variables", "name": "b", "description": "", "templateType": "anything"}, "s3": {"definition": "random(1,-1)", "group": "Ungrouped variables", "name": "s3", "description": "", "templateType": "anything"}, "s5": {"definition": "random(1,-1)", "group": "Ungrouped variables", "name": "s5", "description": "", "templateType": "anything"}, "s1": {"definition": "random(1,-1)", "group": "Ungrouped variables", "name": "s1", "description": "", "templateType": "anything"}, "a": {"definition": "s1*random(2..5)", "group": "Ungrouped variables", "name": "a", "description": "", "templateType": "anything"}, "b1": {"definition": "s3*random(2..9)", "group": "Ungrouped variables", "name": "b1", "description": "", "templateType": "anything"}, "c3": {"definition": "s5*random(2..8)", "group": "Ungrouped variables", "name": "c3", "description": "", "templateType": "anything"}, "a2": {"definition": "s4*random(3..9)", "group": "Ungrouped variables", "name": "a2", "description": "", "templateType": "anything"}, "s4": {"definition": "random(1,-1)", "group": "Ungrouped variables", "name": "s4", "description": "", "templateType": "anything"}, "s2": {"definition": "random(1,-1)", "group": "Ungrouped variables", "name": "s2", "description": "", "templateType": "anything"}}, "metadata": {"notes": "\n \t\t20/06/2012:
\n \t\tAdded tags.
\n \t\tTidied up display of prompt using \\displaystyle.
\n \t\tProblems with display of $e^{ax}$ for $a \\lt 0$. Had brackets around the $a$. (Corrected as an issue 29/06/2012).
\n \t\tMistake in Show steps, corrected.
\n \t\tAdded requirement to enter numbers as fractions or integers.
\n \t\t\n \t\t
3/07/2012:
Added tags.
\n \t\t\n \t\t
9/07/2012:
\n \t\tExtended ruleset std to include !noLeadingMinus so that answer is displayed in the right order.
\n \t\t", "description": "Integrate $f(x) = ae ^ {bx} + c\\sin(dx) + px^q$. Must input $C$ as the constant of integration.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\nIntegrate the following function $f(x)$.
\n
You must input the constant of integration as $C$.