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{"name": "Integration (Negative and Fractional Powers) 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration (Negative and Fractional Powers) 2", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Using the Table of Integrals/Antiderivatives, calculate the integral of $y=\\simplify[unitFactor, unitPower,fractionNumbers]{{a_1}*x^{b_1}+{a_2}*x^{b_2}+{a_3}*x^{b_3}}$.</p>\n<p></p>\n<p></p>", "advice": "<p>From the Table of Integrals&nbsp;we see that a function of the form \\[ f(x)=x^n \\] has the integral \\[ \\int x^n dx&nbsp; =&nbsp; \\frac{x^{n+1}}{n+1}+ c,\\]</p>\n<p>and \\[\\int kf(x) dx = k \\int f(x) dx.\\]</p>\n<p>Additionally, the&nbsp;integral of the sum or difference of two or more functions is equal to the sum or difference of the&nbsp;integrals of each function: \\[ \\int(f(x)\\pm g(x))\\, dx = \\int f(x)\\, dx&nbsp; \\pm \\int g(x) \\, dx.\\]</p>\n<p>So, for the function</p>\n<p>\\[y=\\simplify[all,fractionNumbers]{{a_1}*x^{b_1}+{a_2}*x^{b_2}+{a_3}*x^{b_3}},\\]</p>\n<p>the&nbsp;integral &nbsp;is</p>\n<p>\\[ \\begin{split}\\simplify[all,fractionNumbers]{int({a_1}*x^{b_1}+{a_2}*x^{b_2}+{a_3}*x^{b_3},x)} = \\simplify[all,fractionNumbers]{{a_1}int(x^{b_1},x)+{a_2}int(x^{b_2},x)+{a_3}int(x^{b_3},x)} &amp;\\,=&nbsp;\\simplify[all,fractionNumbers,!collectNumbers,!simplifyFractions,!noLeadingMinus]{{a_1}(x^({b_1+1})/{b_1+1}) +{a_2}(x^({b_2+1})/{b_2+1}) +{a_3}(x^({b_3+1})/{b_3+1})} + c,\\\\ \\\\&amp;\\,= \\simplify[all,fractionNumbers]{{a_1/(b_1+1)}x^{b_1+1}+{a_2/(b_2+1)}x^{b_2+1}+{a_3/(b_3+1)}x^{b_3+1}}+c.\\end{split} \\]</p>\n<p></p>\n<p>Note: You only need to put one $c$ term here, you do&nbsp;<em>not&nbsp;</em>need to put a separate constant term for each calculation.&nbsp;</p>", "rulesets": {}, "extensions": [], "variables": {"a_1": {"name": "a_1", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything"}, "b_1": {"name": "b_1", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)/random(2..10)", "description": "", "templateType": "anything"}, "a_2": {"name": "a_2", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything"}, "b_2": {"name": "b_2", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)/random(2..10)", "description": "", "templateType": "anything"}, "a_3": {"name": "a_3", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything"}, "b_3": {"name": "b_3", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)/random(2..10)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "b_1>b_2 and b_2>b_3\n", "maxRuns": "100"}, "ungrouped_variables": ["a_1", "a_2", "a_3", "b_1", "b_2", "b_3"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a_1}x^{b_1+1}/{b_1+1}+{a_2}x^{b_2+1}/{b_2+1}+{a_3}x^{b_3+1}/{b_3+1}+c", "answerSimplification": "fractionNumbers, basic, unitFactor", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}