// Numbas version: exam_results_page_options
{"name": "Maths Support: Indefinite integration by substitution", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "preventleave": false, "browse": true, "showfrontpage": false, "showresultspage": "never"}, "duration": 0.0, "metadata": {"notes": "", "description": "<p>5 questions on using substitution to find indefinite integrals.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "shufflequestions": false, "questions": [], "percentpass": 50.0, "allQuestions": true, "pickQuestions": 0, "type": "exam", "feedback": {"showtotalmark": true, "advicethreshold": 0.0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Calculus", "Steps", "calculus", "indefinite integration", "integrals", "integration", "integration by substitution", "steps", "substitution"], "advice": "\n    \n    \n    <p>This exercise is best solved by using substitution. <br/>Note that if we let $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$ then $du=\\simplify[std]{({2*a} * x)*dx }$<br/>Hence we can replace $xdx$ by $\\frac{1}{\\var{2*a}}du$.</p>\n    \n    \n    \n    <p>Hence the integral becomes:</p>\n    \n    \n    \n    <p>\\[\\begin{eqnarray*} I&#38;=&#38;\\simplify[std]{Int((1/{2*a})u^{m},u)}\\\\\n    \n    &#38;=&#38;\\simplify[std]{(1/{2*a})u^{m+1}/{m+1}+C}\\\\\n    \n    &#38;=&#38; \\simplify[std]{({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C}\n    \n    \\end{eqnarray*}\\]</p>\n    \n    \n    \n    <p><b>A Useful Result</b><br/>This example can be generalised. <br/>Suppose \\[I = \\int\\; f&#39;(x)g(f(x))\\;dx\\]<br/>The using the substitution $u=f(x)$ we find that $du=f&#39;(x)\\;dx$ and so using the same method as above:<br/>\\[I = \\int g(u)\\;du \\]<br/>And if we can find this simpler integral in terms of $u$ we can replace $u$ by $f(x)$ and get the result in terms of $x$.</p>\n    \n    \n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n      <p>\\[I=\\simplify[std]{Int( x*({a} * x ^ 2 + {b})^{m},x)}\\]</p>\n      <p>$I=\\;$[[0]]</p>\n      <p>Input numbers in your answer as integers or fractions and not as decimals.</p>\n      <p>Click on Show steps to get further help. You will lose 1 mark if you do so.</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Input all numbers as integers or fractions and not as decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C", "type": "jme"}], "steps": [{"prompt": "<p>Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$</p>", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n  <p>Find the following integral.</p>\n  <p>Input the constant of integration as $C$.</p>\n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..5)", "name": "a"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b": {"definition": "s1*random(1..9)", "name": "b"}, "m": {"definition": "random(4..9)", "name": "m"}}, "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>Added decimal point as forbidden string and included message in prompt about not entering decimals.</p>\n    \t\t    <p>Got rid of a redundant ruleset.</p>\n    \t\t    <p>&nbsp;</p>\n    \t\t    <p>&nbsp;</p>\n    \t\t", "description": "<p>Find $\\displaystyle \\int x(a x ^ 2 + b)^{m}\\;dx$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Steps", "calculus", "indefinite integration", "integration", "integration by substitution", "substitution"], "advice": "<p>This exercise is best solved by using substitution.</p>\n  <p>Note that the numerator $\\simplify[std]{{2 * a} * x + {b}}$ of \\[\\simplify[std]{({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c})}\\] is the derivative of the denominator $\\simplify[std]{{a} * x ^ 2 + {b} * x + {c}}$</p>\n  <p>So if you use as your substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$ you then have $\\simplify[std]{ du = ({2 * a} * x + {b}) * dx}$</p>\n  <p>Hence we can replace $\\simplify[std]{ ({2 * a} * x + {b}) * dx}$ by $du$</p>\n  <p>Hence the integral becomes:</p>\n  <p>\\[\\begin{eqnarray*} I&amp;=&amp;\\int\\;\\frac{du}{u}\\\\ &amp;=&amp;\\ln(|u|)+C\\\\ &amp;=&amp; \\simplify[std]{ln(abs({a} * (x ^ 2) + ({b} * x) + {c}))+C} \\end{eqnarray*}\\]<br/><br/></p>\n  <p><b>A Useful Result</b><br/>This example can be generalised. <br/>Suppose \\[I = \\int\\; \\frac{f'(x)}{f(x)}\\;dx\\]<br/>The using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:<br/>\\[I = \\int \\frac{du}{u} = \\ln(|u|)+ C = \\ln(|f(x)|)+C\\]</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n      <p>\\[I=\\simplify[std]{Int(({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c}),x)}\\]</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>Click on Show steps if you need help. You will lose 1 mark if you do so.</p>\n      \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}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "showpreview": true, "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "ln(abs((({a} * (x ^ 2)) + ({b} * x) + {c})))+C", "checkvariablenames": false, "type": "jme"}], "steps": [{"prompt": "<p>Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$</p>", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "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.</p>\n    <p>&nbsp;</p>\n  \n    ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..5)", "name": "a"}, "c": {"definition": "a*b1^2+c1", "name": "c"}, "b": {"definition": "2*a+b1", "name": "b"}, "f": {"definition": "-a*(1+b1)^2", "name": "f"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b1": {"definition": "s1*random(1..5)", "name": "b1"}, "test": {"definition": "4*a*c-b^2", "name": "test"}, "c1": {"definition": "max(-10,f+1)+random(1..5)", "name": "c1"}}, "metadata": {"notes": "\n        \t\t    \t\t<p><strong>2/08/2012:</strong></p>\n        \t\t    \t\t<p>Added tags.</p>\n        \t\t    \t\t<p>Added description.</p>\n        \t\t    \t\t<p>Checked calculation. OK.</p>\n        \t\t    \t\t<p>Added information about Show steps in prompt content area.&nbsp;</p>\n        \t\t    \t\t<p>Added decimal point as forbidden string and included message in prompt about not entering decimals.</p>\n        \t\t    \t\t<p>Got rid of a redundant ruleset.&nbsp;!noLeadingMinus added to std ruleset.</p>\n        \t\t    \t\t<p>Note that the choice of variables means that the argument of the log answer is always $\\gt 0$ so no need to use abs.</p>\n        \t\t    \t\t<p>&nbsp;</p>\n        \t\t    \n        \t\t", "description": "<p>Find $\\displaystyle \\int \\frac{2ax + b}{ax ^ 2 + bx + c}\\;dx$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Calculus", "Steps", "calculus", "constant of integration", "integrals", "integrating trigonometric functions", "integration", "integration by substitution", "steps", "substitution"], "advice": "\n    \n    \n    <p>This exercise is best solved by using substitution. <br/>Note that if we let $u=\\simplify[std]{{a}+{b}cos(x)}$ then $du=\\simplify[std]{({-b}*sin(x))*dx }$<br/>Hence we can replace $\\sin(x)\\;dx$ by $\\frac{1}{\\var{-b}}\\;du$.</p>\n    \n    \n    \n    <p>Hence the integral becomes:</p>\n    \n    \n    \n    <p>\\[\\begin{eqnarray*} I&#38;=&#38;\\simplify[std]{Int((1/{-b})u^{m},u)}\\\\\n    \n    &#38;=&#38;\\simplify[std]{(1/{-b})u^{m+1}/{m+1}+C}\\\\\n    \n    &#38;=&#38; \\simplify[std]{({a}+{b}*cos(x))^{m+1}/{-b*(m+1)}+C}\n    \n    \\end{eqnarray*}\\]</p>\n    \n    \n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n      <p>\\[I=\\simplify[std]{Int( sin(x)*({a} + {b}*cos(x))^{m},x)}\\]</p>\n        <p>Input all numbers as integers or fractions.</p>\n        <p>$I=\\;$[[0]]</p>\n        <p>Input the constant of integration as $C$.</p>\n        <p>Click on Show steps if you need help. You will lose 1 mark if you do so.</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": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "({a}+{b}*cos(x))^{m+1}/{-b*(m+1)}+C", "type": "jme"}], "steps": [{"prompt": "<p>Try the substitution $u=\\simplify[std]{{a}+{b}cos(x)}$</p>", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "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": "random(1..9)", "name": "a"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b": {"definition": "s1*random(1..9)", "name": "b"}, "m": {"definition": "random(3..9)", "name": "m"}}, "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", "description": "<p>Find $\\displaystyle \\int \\sin(x)(a+ b\\cos(x))^{m}\\;dx$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "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"}], "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": []}]}, {"name": "Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "b1"], "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": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": [], "custom_part_types": [], "resources": []}