Do not input numbers as decimals, only as integers without the decimal point, or fractions
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\n$I=\\;$[[0]]
\nInput the constant of integration as $C$.
\nInput all numbers as integers or fractions not as decimals.
\nClick on Show steps if you need help. You will lose 1 mark if you do so.
\n ", "steps": [{"showCorrectAnswer": true, "marks": 0, "scripts": {}, "type": "information", "prompt": "Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$
"}]}], "name": "Luis's copy of Integration: Indefinite integral by substitution", "ungrouped_variables": ["a", "c", "b", "f", "s1", "b1", "c1"], "variables": {"a": {"description": "", "definition": "random(1..5)", "templateType": "anything", "name": "a", "group": "Ungrouped variables"}, "s1": {"description": "", "definition": "random(1,-1)", "templateType": "anything", "name": "s1", "group": "Ungrouped variables"}, "c": {"description": "", "definition": "a*b1^2+c1", "templateType": "anything", "name": "c", "group": "Ungrouped variables"}, "f": {"description": "", "definition": "-a*(1+b1)^2", "templateType": "anything", "name": "f", "group": "Ungrouped variables"}, "b": {"description": "", "definition": "2*a+b1", "templateType": "anything", "name": "b", "group": "Ungrouped variables"}, "c1": {"description": "", "definition": "max(-10,f+1)+random(1..5)", "templateType": "anything", "name": "c1", "group": "Ungrouped variables"}, "b1": {"description": "", "definition": "s1*random(1..5)", "templateType": "anything", "name": "b1", "group": "Ungrouped variables"}}, "preamble": {"css": "", "js": ""}, "statement": "\nFind the following integral.
\nInput the constant of integration as $C$.
\nInput all numbers as integers or fractions.
\n\n ", "metadata": {"description": "
Find $\\displaystyle \\int \\frac{2ax + b}{ax ^ 2 + bx + c}\\;dx$
", "notes": "\n \t\t2/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tChecked calculation. OK.
\n \t\tAdded information about Show steps in prompt content area.
\n \t\tAdded decimal point as forbidden string and included message in prompt about not entering decimals.
\n \t\tGot rid of a redundant ruleset. !noLeadingMinus added to std ruleset.
\n \t\tNote that the choice of variables means that the argument of the log answer is always $\\gt 0$ so no need to use abs.
\n \t\t\n \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "functions": {}, "variable_groups": [], "question_groups": [{"questions": [], "pickQuestions": 0, "name": "", "pickingStrategy": "all-ordered"}], "showQuestionGroupNames": false, "advice": "
This 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*}\\]
\nA Useful Result
This example can be generalised.
Suppose \\[I = \\int\\; \\frac{f'(x)}{f(x)}\\;dx\\]
The 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\\]