// Numbas version: finer_feedback_settings
{"name": "Julie's copy of Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variable_groups": [], "metadata": {"notes": "\n\t\t        \t\t    \t\t<p><strong>2/08/2012:</strong></p>\n\t\t        \t\t    \t\t<p>Added tags.</p>\n\t\t        \t\t    \t\t<p>Added description.</p>\n\t\t        \t\t    \t\t<p>Checked calculation. OK.</p>\n\t\t        \t\t    \t\t<p>Added information about Show steps in prompt content area.&nbsp;</p>\n\t\t        \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    \t\t<p>Got rid of a redundant ruleset.&nbsp;!noLeadingMinus added to std ruleset.</p>\n\t\t        \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    \t\t<p>&nbsp;</p>\n\t\t        \t\t    \n\t\t        \n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find $\\displaystyle \\int \\frac{2ax + b}{ax ^ 2 + bx + c}\\;dx$</p>"}, "showQuestionGroupNames": false, "advice": "\n\t<p>This exercise is best solved by using substitution.</p>\n\t  <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\t  <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\t  <p>Hence we can replace $\\simplify[std]{ ({2 * a} * x + {b}) * dx}$ by $du$</p>\n\t  <p>Hence the integral becomes:</p>\n\t  <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\t  <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>\n\t", "functions": {}, "progress": "ready", "tags": ["Steps", "calculus", "indefinite integration", "integration", "integration by substitution", "substitution"], "variables": {"test": {"definition": "4*a*c-b^2", "name": "test"}, "f": {"definition": "-a*(1+b1)^2", "name": "f"}, "b": {"definition": "2*a+b1", "name": "b"}, "a": {"definition": "random(1..5)", "name": "a"}, "c": {"definition": "a*b1^2+c1", "name": "c"}, "c1": {"definition": "max(-10,f+1)+random(1..5)", "name": "c1"}, "b1": {"definition": "s1*random(1..5)", "name": "b1"}, "s1": {"definition": "random(1,-1)", "name": "s1"}}, "name": "Julie's copy of Indefinite integral by substitution", "parts": [{"prompt": "\n\t\t\t      <p>\\[I=\\simplify[std]{Int(({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c}),x)}\\]</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>Click on Show steps if you need help. You will lose 1 mark if you do so.</p>\n\t\t\t      \n\t\t\t        \n\t\t\t", "stepspenalty": 1.0, "marks": 0.0, "steps": [{"prompt": "<p>Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$</p>", "marks": 0.0, "type": "information"}], "type": "gapfill", "gaps": [{"type": "jme", "checkingaccuracy": 0.001, "showpreview": true, "answersimplification": "std", "answer": "ln(abs((({a} * (x ^ 2)) + ({b} * x) + {c})))+C", "notallowed": {"strings": ["."], "partialcredit": 0.0, "message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions</p>", "showstrings": false}, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "marks": 3.0, "checkingtype": "absdiff", "checkvariablenames": false, "expectedvariablenames": []}]}], "question_groups": [{"pickQuestions": 0, "pickingStrategy": "all-ordered", "name": "", "questions": []}], "type": "question", "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.</p>\n\t    <p>&nbsp;</p>\n\t  \n\t    \n\t", "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/"}]}