// Numbas version: exam_results_page_options
{"name": "Integration: Indefinite integral with 1/ (ax+b)^n", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"js": "", "css": ""}, "metadata": {"notes": "\n\t\t<p><strong>2/08/2012:</strong></p>\n\t\t<p>Added tags.</p>\n\t\t<p>Added description.</p>\n\t\t<p>Added decimal point to forbidden strings along with message to user re input of numbers.</p>\n\t\t<p>Added a Step and message about Show steps included - losing 1 mark if used as it gives the formula for finding the integral. Increased marks to 3 for the question, so that can cope with losing a mark for using Show steps.</p>\n\t\t<p>Changed accuracy setting to relative difference of 0.00001 as we have negative powers.</p>\n\t\t<p>Checked calculation. OK.</p>\n\t\t<p>Added message in prompt &nbsp;about including the constant of integration.</p>\n\t\t<p>Noted issue with steps-answer order and the messages/marks generated.</p>\n\t\t<p>Changed numerator to the range 2..5.</p>\n\t\t<p>Improved display in Advice.</p>\n\t\t<p>&nbsp;</p>\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find $\\displaystyle \\int \\frac{a}{(bx+c)^n}\\;dx$</p>"}, "name": "Integration: Indefinite integral with 1/ (ax+b)^n", "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "marks": 0, "showCorrectAnswer": true, "prompt": "\n\t\t\t<p>$\\displaystyle \\int \\simplify[std]{{b}/(({a}*x+{d})^{n})} dx= \\phantom{{}}$[[0]]</p>\n\t\t\t      <p>Input all numbers as integers or fractions and not decimals. Remember to&nbsp;include the constant of integration $C$.</p>\n\t\t\t      <p>Click on Show steps to get help. You will lose 1 mark by doing so.</p>\n\t\t\t", "gaps": [{"checkingtype": "reldiff", "notallowed": {"message": "<p>Input all numbers as integers or fractions and not decimals.</p>", "strings": ["."], "partialCredit": 0, "showStrings": false}, "vsetrangepoints": 5, "checkvariablenames": false, "answer": "(-{b})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C", "showCorrectAnswer": true, "expectedvariablenames": [], "type": "jme", "scripts": {}, "marks": 3, "checkingaccuracy": 0.0001, "showpreview": true, "answersimplification": "std", "vsetrange": [0, 1]}], "steps": [{"showCorrectAnswer": true, "prompt": "<p>&nbsp;\\[\\int (ax+b)^n \\;dx = \\frac{1}{a(n+1)}(ax+b)^{n+1}+C\\]</p>", "marks": 0, "scripts": {}, "type": "information"}], "type": "gapfill"}], "functions": {}, "advice": "\n\t<p>Let $y = \\simplify[std]{{a}*x+{d}}$. Then,<br/>\\[\\simplify[std]{{b}/(({a}*x+{d})^{n})} = \\simplify[std]{{b}/(y^{n})}.\\]</p>\n\t  <p>Now,<br/>\\[\\int \\simplify[std]{{b}/({a}*x+{d})^{n}} dx = \\int \\simplify[std]{{b}/(y^{n})} \\frac{dx}{dy} dy.\\]</p>\n\t  <p>Rearrange $y = \\simplify[std]{{a}x+{d}}$ to get $\\displaystyle x = \\simplify[std]{(y-{b})/{a}}$, and hence $\\displaystyle\\frac{dx}{dy} = \\frac{1}{\\var{a}}$.</p>\n\t  <p>$\\displaystyle \\int \\frac{1}{y^n} dx = -\\frac{1}{(n-1)y^{n-1}} + C$ is a standard integral, so we can now calculate the desired integral:</p>\n\t  <p>\\[\\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}.\\]</p>\n\t", "type": "question", "variables": {"d": {"templateType": "anything", "definition": "random(1..9)", "name": "d", "group": "Ungrouped variables", "description": ""}, "n": {"templateType": "anything", "definition": "random(3..5)", "name": "n", "group": "Ungrouped variables", "description": ""}, "a": {"templateType": "anything", "definition": "random(2..9)", "name": "a", "group": "Ungrouped variables", "description": ""}, "b": {"templateType": "anything", "definition": "random(2..5)", "name": "b", "group": "Ungrouped variables", "description": ""}}, "variable_groups": [], "ungrouped_variables": ["a", "b", "d", "n"], "tags": ["calculus", "Calculus", "checked2015", "constant of integration", "indefinite integration", "integration", "integration by substitution", "MAS1601", "mas1601", "standard integrals", "Steps", "steps", "substitution"], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\n\t    \n\t    \n\t    <p>Find the following indefinite integral.</p>\n\t    \n\t    \n\t    \n\t    <p>Input the constant of integration as $C$.</p>\n\t    \n\t    \n\t", "question_groups": [{"pickQuestions": 0, "name": "", "questions": [], "pickingStrategy": "all-ordered"}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Johnny Yi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2810/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Johnny Yi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2810/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}