// Numbas version: exam_results_page_options
{"name": "Patrick's copy of Integration by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"a": {"group": "Ungrouped variables", "description": "", "definition": "random(2..6)", "templateType": "anything", "name": "a"}, "b": {"group": "Ungrouped variables", "description": "", "definition": "s1*random(1..9)", "templateType": "anything", "name": "b"}, "c": {"group": "Ungrouped variables", "description": "", "definition": "ceil(b^2/4a+random(1..5))", "templateType": "anything", "name": "c"}, "s1": {"group": "Ungrouped variables", "description": "", "definition": "random(1,-1)", "templateType": "anything", "name": "s1"}}, "parts": [{"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 all numbers as integers or fractions.</p>\n      <p>Do not forget to include the constant of integration $C$.</p>\n      ", "steps": [{"prompt": "\n          <p>Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$</p>\n          <p>Note that \\[\\int \\frac{1}{x}\\;dx=\\ln(|x|)+C\\] and you must input the absolute value of the argument of the natural logarithm. You input the absolute value using abs, for example abs(x)=$\\simplify{abs(x)}$</p>\n          ", "type": "information", "showFeedbackIcon": true, "marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}}], "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "gaps": [{"showpreview": true, "showFeedbackIcon": true, "answersimplification": "std", "vsetrange": [0, 1], "notallowed": {"message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "checkingtype": "absdiff", "answer": "ln(abs((({a} * (x ^ 2)) + ({b} * x) + {c})))+C", "expectedvariablenames": [], "musthave": {"message": "<p>Note that \\[\\int \\frac{1}{x}\\;dx=\\ln(|x|)+C\\] and you must input the absolute value of the argument of the natural logarithm. You input the absolute value using abs, for example abs(x)=$\\simplify{abs(x)}$</p>", "showStrings": false, "partialCredit": 0, "strings": ["abs"]}, "vsetrangepoints": 5, "type": "jme", "marks": 3, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "checkingaccuracy": 0.001, "checkvariablenames": false}], "stepsPenalty": 1}], "extensions": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variable_groups": [], "functions": {}, "advice": "\n  <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><strong>A Useful Result</strong><br/>This example can be generalised.</p>\n  <p>Suppose \\[I = \\int\\; \\frac{f'(x)}{f(x)}\\;dx\\]</p>\n  <p>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  ", "statement": "<p>Find the following integral.</p>\n<p>You must input the constant of integration as $C$.</p>\n<p>Input all numbers as integers or fractions.</p>\n<p>You can click on Show steps to get a hint. You will lose 1 mark if you do so.</p>\n<p>Note that $\\displaystyle \\int \\frac{1}{x}\\;dx=\\ln(|x|)+C$ and you must include the absolute value in the argument of $\\ln$. You input $|x|$ as abs(x).</p>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find $\\displaystyle I=\\int \\frac{2 a x + b} {a x ^ 2 + b x + c}\\;dx$ by substitution or otherwise.</p>"}, "preamble": {"js": "", "css": ""}, "tags": [], "ungrouped_variables": ["a", "c", "b", "s1"], "name": "Patrick's copy of Integration by substitution", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "type": "question", "contributors": [{"name": "Patrick Joyce", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1674/"}]}]}], "contributors": [{"name": "Patrick Joyce", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1674/"}]}