// 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": {}, "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"], "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": 1.0, "prompt": "\n      <p>\\[I=\\int\\frac{\\simplify[std]{{a}*x+{b}}}{(1-x^2)^{1/2}} \\;dx\\]</p>\n        <p>$I=\\;$[[0]]</p>\n        <p>Input the constant of integration as $C$.</p>\n        <p>Input all numbers as integers or fractions not as decimals.</p>\n        <p>&nbsp;</p>\n        <p>Click on Show steps if you need help. You will lose 1 mark if you do so.</p>\n        <p>&nbsp;</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.0001, "vsetrange": [0.0, 0.9], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "-{a}*(1-x^2)^(1/2)+{b}*arcsin(x)+C", "type": "jme"}], "steps": [{"prompt": "\n          <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            <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          ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "extensions": [], "statement": "\n  <p>Find the following integral.</p>\n    <p>Input the constant of integration as $C$.</p>\n    <p>Input all numbers as integers or fractions not as decimals.</p>\n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "2*s1*random(1..5)", "name": "a"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b": {"definition": "2*random(1..5)+random(1,-1)", "name": "b"}}, "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>Got rid of a redundant ruleset.&nbsp;!noLeadingMinus added to std ruleset.</p>\n    \t\t<p>Changed from $\\sqrt{1-x^2}$ to $(1-x^2)^{1/2}$ throughout as display was not good.</p>\n    \t\t<p>&nbsp;</p>\n    \t\t", "description": "<p>Find $\\displaystyle \\int\\frac{ax+b}{(1-x^2)^{1/2}} \\;dx$. Solution involves inverse trigonometric functions.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "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/"}]}