// Numbas version: finer_feedback_settings
{"name": "Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "c", "b", "b1"], "name": "Indefinite integral by substitution", "tags": ["Calculus", "Steps", "arcsin", "calculus", "constant of integration", "integrals", "integration", "integration by substitution", "inverse trigonometric functions", "standard integrals", "steps", "substitution"], "preamble": {"css": "", "js": ""}, "advice": "\n  <p>For the integral \\[I=\\simplify[std]{Int((({c}) / (sqrt({a}-{b}x^2))),x)}\\] use the substitution $\\displaystyle \\simplify[std]{u=(sqrt({b})/sqrt({a}))*x}$<br/>so that \\[\\simplify[all,!sqrtProduct,fractionNumbers]{sqrt({a}-{b}x^2)=sqrt({a}-{b}*({a}/{b})*u^2)=sqrt({a}-{a}*u^2)=sqrt({a})*sqrt(1-u^2)}\\]</p>\n    <p>We have $\\displaystyle \\simplify[std]{du=(sqrt({b})/sqrt({a}))dx}$ and we get<br/>\\[\\begin{eqnarray*}I&amp;=&amp;\\simplify[std]{({c}*(sqrt({a})/sqrt({b})))*Int((1 / ( sqrt({a})*sqrt(1-u^2)&nbsp;)),u)}\\\\ &nbsp;&amp;=&amp;\\simplify[std]{({c}/sqrt({b}))*Int((1 / (sqrt(1-u^2))),u)}\\\\ &amp;=&amp;\\simplify[std]{({c}/sqrt({b}))*arcsin(u)+C}\\\\ &amp;=&amp;\\simplify[std]{({c}/sqrt({b}))*arcsin((sqrt({b})/sqrt({a}))*x)+C} \\end{eqnarray*}\\]<br/>on replacing $u$ by $\\displaystyle \\simplify[std]{(sqrt({b})/sqrt({a}))*x}$</p>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "parts": [{"stepsPenalty": 1, "prompt": "<p>\\[I=\\simplify[std]{Int(({c} / (sqrt({a}-{b}x^2))),x)}\\]</p>\n<p>$I=\\;$[[0]]</p>\n<p>Input all numbers as integers, fractions or surds. No decimal numbers. You input surds, for example $\\sqrt{2}$, by writing <code>sqrt(2)</code>.</p>\n<p>Input the constant of integration as $C$.</p>\n<p>You can get help by clicking on <em>Show steps</em>. You will lose 1 mark if you do so.</p>", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions or surds (such as sqrt(2) for $\\sqrt{2}$).</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 0.25], "showpreview": true, "marks": 3, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({c}/sqrt({b}))*arcsin((sqrt({b})/sqrt({a}))*x)+C", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "steps": [{"showCorrectAnswer": true, "prompt": "<p>Try the substitution $\\displaystyle \\simplify[std]{u=(sqrt({b})/sqrt({a}))*x}$ and then consider the standard integral \\[\\int \\frac{dx}{\\sqrt{1-x^2}}=\\arcsin(x)+C\\]</p>", "scripts": {}, "type": "information", "marks": 0}], "type": "gapfill"}], "statement": "\n  <p>Find the following integral.</p>\n    <p>&nbsp;</p>\n  ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "if(b1=a,b1+1,b1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "b1": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}}, "metadata": {"notes": "\n    \t\t<p><strong>2/08/2012:</strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p>Added description.</p>\n    \t\t<p>Checked calculation. OK.</p>\n    \t\t<p>Added information about Show steps in prompt content area.&nbsp;</p>\n    \t\t<p>Corrected error in Show steps, the substitution was the wrong way round.</p>\n    \t\t<p>Simplified the presentation of Advice.</p>\n    \t\t", "description": "<p>Find $\\displaystyle \\int \\frac{c}{\\sqrt{a-bx^2}}\\;dx$. Solution involves the inverse trigonometric function $\\arcsin$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}