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{"name": "Integration: By Parts 6", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration: By Parts 6", "tags": [], "metadata": {"description": "<p>Calculating the integral of a function of the form $e^{ax} \\cos(x)$ using integration by parts.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Calculate the integral \\[ \\simplify{int(e^({a}x) cos(x),x)}\\]</p>", "advice": "<p>If we have a function of $x$ which is the product of two functions of $x$, to integrate such a function it is often necessary to use Integration by Parts. The formula for Integration by Parts is:</p>\n<p>\\[ \\int u(x) \\frac{dv}{dx} dx = u(x)v(x) - \\int v(x) \\frac{du}{dx} dx.\\]</p>\n<p>Using this method can be broken down into steps:</p>\n<ol>\n<li>Identify $u(x)$ and $\\tfrac{dv}{dx}$ (The function you pick for each is important, in general you want $u(x)$ to become simpler when differentiating it, and you must be able to integrate $\\tfrac{dv}{dx}$ to find $v(x)$);</li>\n<li>Calculate $\\tfrac{du}{dx}$ and $v(x)$;</li>\n<li>Put the functions $u(x)$, $v(x)$, and their derivatives into the Integration by Parts formula;</li>\n<li>Calculate the integral $\\int v(x) \\tfrac{du}{dx} dx$ (This may require you to use Integration by Parts again, this is OK!);</li>\n<li>Simplify your answer where possible and don't forget to add the constant of integration.</li>\n</ol>\n<p></p>\n<p>For the integral</p>\n<p>\\[ \\simplify{int(e^({a}x)cos(x),x)},\\]</p>\n<p>we must first identify $u(x)$ and $\\tfrac{dv}{dx}$. In this case, let \\[ u(x)=\\simplify{e^({a}x)},\\quad \\frac{dv}{dx}= \\simplify{cos(x)}. \\]</p>\n<p>Next, we need to calculate $\\tfrac{du}{dx}$ and $v(x)$:</p>\n<p>\\[ \\begin{split} u(x) =&nbsp;\\simplify{e^({a}x)} \\quad &amp;\\implies \\frac{du}{dx} = \\simplify{{a}e^({a}x)}; \\\\ \\frac{dv}{dx} = \\cos(x) &amp;\\implies v(x) = \\sin(x). \\end{split} \\]</p>\n<p>Plugging these 4 terms into the integration by parts formula:</p>\n<p>\\[&nbsp; \\begin{split} \\simplify{int(e^({a}x)cos(x),x)} &amp;\\,= \\simplify{e^({a}x)sin(x) - int({a}e^({a}x)sin(x),x)}, \\\\ &amp;\\,= \\simplify{e^({a}x)sin(x)-{a}int(e^({a}x)sin(x),x)}.\\end{split} \\]</p>\n<p>Since the integral on the right-hand side is still the product of two functions of $x$, we need to use integration by parts again.&nbsp;</p>\n<p>So, for&nbsp;</p>\n<p>\\[ \\simplify{int(e^({a}x)sin(x),x)}, \\]</p>\n<p>&nbsp;Let $u=\\simplify{e^({a}x)}$ and $\\tfrac{dv}{dx} = \\sin(x)$. Therefore, $\\tfrac{du}{dx}=\\simplify{{a}e^({a}x)}$ and $v(x)=\\simplify{-cos(x)}$.</p>\n<p>Hence,</p>\n<p>\\[ \\begin{split} \\simplify{int(e^({a}x)sin(x),x)} &amp;\\,= \\simplify{-e^({a}x)cos(x)- int(-{a}e^({a}x)cos(x),x)} \\\\ \\\\ &amp;\\,= \\simplify[!noLeadingMinus]{-e^({a}x)cos(x)+{a} int(e^({a}x)cos(x),x)}. \\end{split}\\]</p>\n<p>Plugging this back into the original calculation:</p>\n<p>\\[&nbsp; \\begin{split} \\simplify{int(e^({a}x)cos(x),x)} &amp;\\,=&nbsp;\\simplify{e^({a}x)sin(x)-{a}int(e^({a}x)sin(x),x)},\\\\ \\\\&amp;\\,=\\simplify[!noLeadingMinus]{e^({a}x)sin(x)-{a}[-e^({a}x)cos(x)+{a} int(e^({a}x)cos(x),x)]}, \\\\&amp;\\,=\\simplify[!noLeadingMinus]{e^({a}x)sin(x)+{a}e^({a}x)cos(x)-{a^2} int(e^({a}x)cos(x),x)}. \\end{split} \\]</p>\n<p>At this point it seems like we may have made a mistake since we&nbsp;<em>still</em> have an interal on the right-hand side made up of two functions of $x$. However, this is&nbsp;<em>a multiple</em>&nbsp;of<em>&nbsp;the initial integral.&nbsp;</em></p>\n<p>So, rather than trying to do integration by parts again (which won't work!), we can combine the two integral terms together on the left-hand side:</p>\n<p>\\[&nbsp; \\begin{split} \\simplify{int(e^({a}x)cos(x),x)} &amp;\\,=&nbsp;\\simplify[!noLeadingMinus]{e^({a}x)sin(x)+{a}e^({a}x)cos(x)-{a^2} int(e^({a}x)cos(x),x)}, \\\\ \\simplify{{1+a^2}int(e^({a}x)cos(x),x)} &amp;\\,=&nbsp;\\simplify{e^({a}x)sin(x)+{a}e^({a}x)cos(x)} .\\end{split} \\]</p>\n<p>Therefore,</p>\n<p>\\[ \\begin{split} \\simplify{int(e^({a}x)cos(x),x)} &amp;\\,=&nbsp;\\simplify{1/{1+a^2}(e^({a}x)sin(x)+{a}e^({a}x)cos(x))}+c, \\\\ &amp;\\,=&nbsp;\\simplify{1/{1+a^2}e^({a}x)(sin(x)+{a}cos(x))}+c. \\end{split} \\]</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..4)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a"], "variable_groups": [], "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": "1/{1+a^2}e^({a}x)(sin(x)+{a}cos(x))+c", "answerSimplification": "fractionNumbers, basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "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/"}]}