// 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", "s1", "b"], "name": "Indefinite integral by substitution", "tags": ["arcsin", "Calculus", "calculus", "constant of integration", "integrals", "integration", "integration by substitution", "inverse trigonometric functions", "rebel", "REBEL", "Rebel", "rebelmaths", "standard integrals", "Steps", "steps", "substitution"], "advice": "<p>Split the integral into two parts<br/>\\[I=\\int\\frac{\\simplify[std]{{a}*x}}{(1-x^2)^{1/2}} \\;dx+\\int\\frac{\\simplify[std]{{b}}}{(1-x^2)^{1/2}} \\;dx\\]&nbsp;&nbsp;<br/>For the integral \\[I_1=\\int\\frac{\\simplify[std]{{a}*x}}{(1-x^2)^{1/2}} \\;dx&nbsp;\\] use the substitution $u=1-x^2$ and then $du=-2xdx$ and we get<br/>\\[\\begin{eqnarray*}I_1&amp;=&amp;\\simplify[std]{{-a}/2*Int((1 / (u^(1/2))),u)}\\\\\\\\ &amp;=&amp;\\simplify[std]{({-a}/2)*(2u^(1/2))+C}\\\\ &amp;=&amp;\\simplify[std]{({-a})*(1-x^2)^(1/2)+C} \\end{eqnarray*}\\]<br/>The other integral is a standard result: \\[I_2=\\simplify[std]{Int((({b}) / (1-x^2)^(1/2)),x)={b}*arcsin(x)+C}\\]<br/>Putting these together gives:<br/>\\[I=I_1+I_2=\\simplify[std]{-{a}*(1-x^2)^(1/2)+{b}*arcsin(x)+C}\\]</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 1, "prompt": "\n\t\t\t      <p>\\[I=\\int\\frac{\\simplify[std]{{a}*x+{b}}}{(1-x^2)^{1/2}} \\;dx\\]</p>\n\t\t\t        <p>$I=\\;$[[0]]</p>\n\t\t\t        <p>Input the constant of integration as $C$.</p>\n\t\t\t        <p>Input all numbers as integers or fractions not as decimals.</p>\n\t\t\t        <p>&nbsp;</p>\n\t\t\t        <p>Click on Show steps if you need help. You will lose 1 mark if you do so.</p>\n\t\t\t        <p>&nbsp;</p>\n\t\t\t      \n\t\t\t", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": [], "checkingaccuracy": 0.0001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "-{a}*(1-x^2)^(1/2)+{b}*arcsin(x)+C", "marks": 3, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 0.9]}], "steps": [{"prompt": "\n\t\t\t\t\t          <p>Split the integral into two parts:<br/>\\[I=\\int\\frac{\\simplify[std]{{a}*x}}{(1-x^2)^{1/2}} \\;dx+\\int\\frac{\\simplify[std]{{b}}}{(1-x^2)^{1/2}} \\;dx\\]&nbsp;</p>\n\t\t\t\t\t            <p>Try the substitution $u=1-x^2$ for the first integral and the second one is a standard integral i.e. \\[\\int \\frac{dx}{(1-x^2)^{1/2}}=\\arcsin(x)+C\\]</p>\n\t\t\t\t\t          \n\t\t\t\t\t", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill"}], "extensions": [], "statement": "\n\t  <p>Find the following integral.</p>\n\t    <p>Input the constant of integration as $C$.</p>\n\t    <p>Input all numbers as integers or fractions not as decimals.</p>\n\t  \n\t", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "2*s1*random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "b": {"definition": "2*random(1..5)+random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}}, "metadata": {"description": "<p>Find $\\displaystyle \\int\\frac{ax+b}{(1-x^2)^{1/2}} \\;dx$. Solution involves inverse trigonometric functions.</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}]}