Find the following integral.
\nYou must input the constant of integration as $C$.
\nInput all numbers as integers or fractions.
\nYou can click on Show steps to get a hint. You will lose 1 mark if you do so.
\nNote 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).
\n ", "name": "Luis's copy of Integration by substitution", "metadata": {"notes": "\n \t\t20/06/2012:
\n \t\tAdded tags.
\n \t\tExtensively changed the variables in order to simplify the question - quadratics generated which only have complex roots to stop the quadratic being zero in any range.
\n \t\tAdded Steps prompt about losing a mark. Also added to Steps re using abs.
\n \t\tAdded required string abs for answer entry.
\n \t\tImproved spacing.
\n \t\t4/07/2012:
In the steps it is explained that the absolute value of the natural logarithm is inputted as ln(abs(x)) however unless steps are revealed it is unclear how to write |x|. Should this information be included in the question somehow?
\n \t\t6/08/2012:
\n \t\tInformation on using ln(abs(x)) included in the statement.
\n \t\t\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "
Find $\\displaystyle I=\\int \\frac{2 a x + b} {a x ^ 2 + b x + c}\\;dx$ by substitution or otherwise.
"}, "ungrouped_variables": ["a", "c", "b", "s1"], "parts": [{"prompt": "\n\\[I=\\simplify[std]{Int(({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c}),x)}\\]
\n$I=\\;$[[0]]
\nInput all numbers as integers or fractions.
\nDo not forget to include the constant of integration $C$.
\n ", "type": "gapfill", "marks": 0, "showCorrectAnswer": true, "gaps": [{"checkvariablenames": false, "showCorrectAnswer": true, "answer": "ln(abs((({a} * (x ^ 2)) + ({b} * x) + {c})))+C", "vsetrange": [0, 1], "answersimplification": "std", "expectedvariablenames": [], "musthave": {"partialCredit": 0, "message": "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)=$\\simplify{abs(x)}$
", "showStrings": false, "strings": ["abs"]}, "vsetrangepoints": 5, "type": "jme", "marks": 3, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "scripts": {}, "notallowed": {"partialCredit": 0, "message": "Do not input numbers as decimals, only as integers without the decimal point, or fractions
", "showStrings": false, "strings": ["."]}, "showpreview": true}], "stepsPenalty": 1, "scripts": {}, "steps": [{"prompt": "\nTry the substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$
\nNote 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)=$\\simplify{abs(x)}$
\n ", "type": "information", "marks": 0, "scripts": {}, "showCorrectAnswer": true}]}], "variables": {"s1": {"definition": "random(1,-1)", "name": "s1", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b": {"definition": "s1*random(1..9)", "name": "b", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "c": {"definition": "ceil(b^2/4a+random(1..5))", "name": "c", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a": {"definition": "random(2..6)", "name": "a", "templateType": "anything", "group": "Ungrouped variables", "description": ""}}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "type": "question", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "advice": "\nThis exercise is best solved by using substitution.
\nNote 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 $\\simplify[std]{{a} * x ^ 2 + {b} * x + {c}}$
\nSo if you use as your substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$ you then have $\\simplify[std]{ du = ({2 * a} * x + {b}) * dx}$
\nHence we can replace $\\simplify[std]{ ({2 * a} * x + {b}) * dx}$ by $du$
\nHence the integral becomes:
\n\\[\\begin{eqnarray*} I&=&\\int\\;\\frac{du}{u}\\\\ &=&\\ln(|u|)+C\\\\ &=& \\simplify[std]{ln(abs({a} * (x ^ 2) + ({b} * x) + {c}))+C} \\end{eqnarray*}\\]
A Useful Result
This example can be generalised.
Suppose \\[I = \\int\\; \\frac{f'(x)}{f(x)}\\;dx\\]
\nThe using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:
\\[I = \\int \\frac{du}{u} = \\ln(|u|)+ C = \\ln(|f(x)|)+C\\]