Find $\\displaystyle \\int \\frac{a}{(bx+c)^n}\\;dx$
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Find the following indefinite integral.
\nInput the constant of integration as $C$.
", "advice": "\n\tLet $y = \\simplify[std]{{a}*x+{d}}$. Then,
\\[\\simplify[std]{{b}/(({a}*x+{d})^{n})} = \\simplify[std]{{b}/(y^{n})}.\\]
Now,
\\[\\int \\simplify[std]{{b}/({a}*x+{d})^{n}} dx = \\int \\simplify[std]{{b}/(y^{n})} \\frac{dx}{dy} dy.\\]
Rearrange $y = \\simplify[std]{{a}x+{d}}$ to get $\\displaystyle x = \\simplify[std]{(y-{b})/{a}}$, and hence $\\displaystyle\\frac{dx}{dy} = \\frac{1}{\\var{a}}$.
\n\t$\\displaystyle \\int \\frac{1}{y^n} dx = -\\frac{1}{(n-1)y^{n-1}} + C$ is a standard integral, so we can now calculate the desired integral:
\n\t\\[\\int \\simplify[std]{{b}/(y^{n})} \\frac{dx}{dy} dy = \\simplify[std]{{b}/({n-1}*y^{n-1})} \\cdot \\frac{1}{\\var{a}} + C = \\simplify[std]{(-{b})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C}.\\]
\n\t \n\t", "rulesets": {"surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}], "std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "extensions": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["n", "b", "a", "d"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "\n\t\t\t$\\displaystyle \\int \\simplify[std]{{b}/(({a}*x+{d})^{n})} dx= \\phantom{{}}$[[0]]
\n\t\t\tInput all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.
\n\t\t\tClick on Show steps to get help. You will lose 1 mark by doing so.
\n\t\t\t \n\t\t\t", "stepsPenalty": 1, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "\\[\\int (ax+b)^n \\;dx = \\frac{1}{a(n+1)}(ax+b)^{n+1}+C\\]
"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 3, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "malrules:\n [\n [\"{-b}/({a*(n-1)}*({a}*x+{d})^{n-1})\", \"Almost there! Did you forget to include the integration constant?\"],\n [\"{-b}/({n-1}*({a}*x+{d})^{n-1})+C\", \"You are on the right track but it looks like you forgot to divide by the number in front of $x$. If you used substitution, double check the relationship between $du$ and $dx$.\"],\n [\"{-b}/({n-1}*({a}*x+{d})^{n-1})\", \"You are on the right track but it looks like you forgot to divide by the number in front of $x$. If you used substitution, double check the relationship between $du$ and $dx$.\"],\n [\"{b}/{a}*ln(({a}*x+{d})^{n})+C\", \"$\\\\int \\\\frac{1}{x} dx = \\\\ln x$. However, you can only use the $\\\\ln$ rule if there is a linear term (without a power) below the line i.e. $ax+b$ but not $ax^n+b$ or $(ax+b)^n$ for $n \\\\neq 1$ etc.\"],\n [\"{b}/{a}*ln(({a}*x+{d})^{n})\", \"$\\\\int \\\\frac{1}{x} dx = \\\\ln x$. However, you can only use the $\\\\ln$ rule if there is a linear term (without a power) below the line i.e. $ax+b$ but not $ax^n+b$ or $(ax+b)^n$ for $n \\\\neq 1$ etc.\"],\n [\"{b}*ln(({a}*x+{d})^{n})+C\", \"$\\\\int \\\\frac{1}{x} dx = \\\\ln x$. However, you can only use the $\\\\ln$ rule if there is a linear term (without a power) below the line i.e. $ax+b$ but not $ax^n+b$ or $(ax+b)^n$ for $n \\\\neq 1$ etc.\"],\n [\"{b}*ln(({a}*x+{d})^{n})\", \"$\\\\int \\\\frac{1}{x} dx = \\\\ln x$. However, you can only use the $\\\\ln$ rule if there is a linear term (without a power) below the line i.e. $ax+b$ but not $ax^n+b$ or $(ax+b)^n$ for $n \\\\neq 1$ etc.\"],\n [\"-1/({a*(n-1)}*({a}*x+{d})^{n-1})+C\", \"Don't forget about the number that was above the line!\"],\n [\"-1/({a*(n-1)}*({a}*x+{d})^{n-1})\", \"Don't forget about the number that was above the line!\"]\n ]\n\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))