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{"name": "Integration: By Substitution 1c", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration: By Substitution 1c", "tags": [], "metadata": {"description": "<p>Calculating the integral of a function of the form $x(ax^2+b)^n$ using integration by substitution.</p>", "licence": "None specified"}, "statement": "<p>Calculate \\[ \\simplify{int(x({a}x^2+{b})^{n},x)}\\]</p>", "advice": "<p>Since this integral is of the form \\[ \\int a \\,g'(x)f(g(x))\\,dx,\\] where $a$ is a constant, we can use the method of substitution to solve it.&nbsp;</p>\n<p>Firstly, we must make a change of variables from $x$ to $u$, where $u$ is equal to the 'inner' function, $g(x)$.</p>\n<p>So, for \\[ \\simplify{int(x({a}x^2+{b})^{n},x)},\\]</p>\n<p>let&nbsp; $\\color{red}{u=\\simplify{{a}x^2+{b}}}.$</p>\n<p>Now, we need to calculate the differential, $du$, where \\[ du = \\left(\\frac{du}{dx}\\right)dx. \\]</p>\n<p>Differentiating $u$ with respect to $x$:</p>\n<p>\\[ \\frac{du}{dx}= \\simplify{{2a}x}.\\]</p>\n<p>Therefore, \\[ \\color{blue}{du = \\simplify{{2a}x}\\, dx}.\\]</p>\n<p>We can see that rather than having the&nbsp;<em>exact</em> expression for the differential in the integral, we have a fractional multiple of it instead. It is good practice to write the integral again so that it does include the differential, and&nbsp;the multiple&nbsp;<em>on the outside</em>. This&nbsp;can make it easier when expressing it in terms of $u$:</p>\n<p>\\[ \\simplify{int(x({a}x^2+{b})^{n},x)} = \\frac{1}{\\var{2a}}\\int\\color{blue}{\\simplify{{2a}x}}(\\color{red}{\\simplify{{a}x^2+{b}}})^\\var{n} \\color{blue}{\\text{d}x}=\\frac{1}{\\var{2a}} \\int \\color{red}{u}^\\var{n} \\color{blue}{\\text{d}u}.\\]</p>\n<p>Calculating the integral in terms of $u$:</p>\n<p>\\[ \\begin{split} \\simplify[fractionNumbers]{(1/{2a}) int(u^{n},u)} &amp;\\,=\\frac{1}{\\var{2a}} \\left(\\simplify{( u^{n+1}/{n+1})}\\right) + c &amp;\\,=\\simplify[fractionNumbers]{{1/(2a*(n+1))} u^{n+1} +c}. \\end{split}\\]</p>\n<p>Finally,&nbsp;rewriting&nbsp;the solution back in terms of the original variable $x$:</p>\n<p>\\[ \\simplify[fractionNumbers]{{1/(2a*(n+1))} u^{n+1} + c = {1/(2a*(n+1))} ({a}x^2+{b})^{n+1} + c}.\\]</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-8..8 except 0)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": ["a", "b", "n"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({a}x^2+{b})^{n+1}/{2a*(n+1)}+c", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}