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{"name": "Integration: Definite Integrals 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration: Definite Integrals 3", "tags": [], "metadata": {"description": "<p>Calculating the definite integral $\\int_{n_1}^{n_2}a \\sin(bx) dx$, where $n_1$ and $n_2$ are multiples of $\\frac{\\pi}{12}$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Evaluate \\[ \\int_\\simplify{{n_1}pi/12}^\\simplify{{n_2}pi/12}\\simplify{{a}sin({b}x)}dx\\]</p>", "advice": "<p>Integrating a function of the form&nbsp;&nbsp;\\[ f(x)=\\sin(kx) \\] has the integral \\[ \\int_a^b&nbsp;\\sin(kx)&nbsp;dx&nbsp; =&nbsp; \\left[-\\frac{1}{k}\\cos(kx)\\right]_a^b,\\]</p>\n<p>and \\[\\int_a^b kf(x) dx = k \\int_a^b f(x) dx.\\]</p>\n<p></p>\n<p>Therefore,</p>\n<p>\\[ \\begin{split} \\simplify{defint({a}sin({b}x),x,{n_1}pi/12,{n_2}pi/12)} &amp;\\,=&nbsp;\\simplify{{a}defint(sin({b}x),x,{n_1}pi/12,{n_2}pi/12)}, \\\\ &amp;\\,= \\left[ \\simplify{-{a}/{b}cos({b}x)}\\right]_\\simplify{{n_1}pi/12}^\\simplify{{n_2}pi/12}, \\\\ &amp;\\,=&nbsp;\\left[\\simplify[all,!trig,fractionNumbers]{-{a/b}cos({simp2}pi)}\\right]-\\left[\\simplify[all,!trig,fractionNumbers]{-{a/b} cos({simp1}pi)}\\right] \\\\ &amp;\\,=\\left[\\simplify[all,fractionNumbers]{-{a*frac2/b}}\\right]-\\left[\\simplify[all,fractionNumbers]{-{a*frac1/b}}\\right]\\\\ &amp;\\,=\\simplify[all,fractionNumbers]{{(-a*frac2+a*frac1)/b}} \\end{split} \\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2,3)", "description": "", "templateType": "anything", "can_override": false}, "n_1": {"name": "n_1", "group": "Ungrouped variables", "definition": "random([2,3,4,6,12])", "description": "", "templateType": "anything", "can_override": false}, "n_2": {"name": "n_2", "group": "Ungrouped variables", "definition": "random(random([2,3,4,6,12,24]))", "description": "", "templateType": "anything", "can_override": false}, "simp1": {"name": "simp1", "group": "Ungrouped variables", "definition": "b/12 * n_1", "description": "", "templateType": "anything", "can_override": false}, "simp2": {"name": "simp2", "group": "Ungrouped variables", "definition": "(b/12)*n_2", "description": "", "templateType": "anything", "can_override": false}, "frac1": {"name": "frac1", "group": "Ungrouped variables", "definition": "cos(simp1*pi)", "description": "", "templateType": "anything", "can_override": false}, "frac2": {"name": "frac2", "group": "Ungrouped variables", "definition": "cos(simp2*pi)", "description": "", "templateType": "anything", "can_override": false}, "sol": {"name": "sol", "group": "Ungrouped variables", "definition": "precround({a*(frac1-frac2)/b},3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "n_1<n_2 and ((b*n_1)/12 < 3/4 or (b*n_1)/12 > 3/4) and ((b*n_2)/12 < 3/4 or (b*n_2)/12 > 3/4)", "maxRuns": "100"}, "ungrouped_variables": ["a", "b", "n_1", "n_2", "simp1", "simp2", "frac1", "frac2", "sol"], "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]] (<em>Give your answer as a fraction or to 2 decimal places, where necessary)</em></p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "answer": "sol", "answerSimplification": "fractionNumbers, basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], 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