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{"name": "Integration: Using a Table of Integrals 6", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration: Using a Table of Integrals 6", "tags": [], "metadata": {"description": "<p>Calculating the integral of a function of the form $\\frac{a}{\\sqrt{b^2-x^2}}$ using a table of integrals.</p>", "licence": "None specified"}, "statement": "<p>Using the Table of Integrals/Antiderivatives, calculate the&nbsp;integral of $f(x)=\\simplify[unitFactor]{{a}/(sqrt({b^2}-x^2))}.$</p>\n<p></p>", "advice": "<p>From the Table of Integrals&nbsp;we see that a function of the form \\[f(x)=\\simplify[unitFactor]{1/(sqrt(a^2-x^2))}\\] has the integral \\[ \\int\\simplify[unitFactor]{1/(sqrt(a^2-x^2))} \\,dx&nbsp; = \\simplify{arcsin(x/a)+c},\\]</p>\n<p><strong>(Reminder: <em>$\\arcsin(x) \\equiv \\sin^{-1}(x)$</em>)</strong></p>\n<p>and \\[ \\int kf(x) \\,dx = k \\int f(x) \\, dx.\\]</p>\n<p>So, for the function</p>\n<p>\\[f(x)=\\simplify[unitFactor]{{a}/(sqrt({b^2}-x^2))},\\]</p>\n<p>the integral is</p>\n<p>\\[ \\int\\simplify[unitFactor]{{a}/(sqrt({b^2}-x^2))} \\,dx \\,=\\simplify[unitFactor]{{a}int(1/(sqrt({b^2}-x^2)),x)} \\,=\\simplify[all]{{a}(arcsin(x/{b}))} +c.&nbsp; \\]</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(1..9)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": ["a", "b"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "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": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a}arcsin(x/{b})+c", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": 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/"}]}