Find $\\displaystyle \\int ae ^ {bx}+ c\\sin(dx) + px ^ {q} + k/x \\;dx$.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\n\tIntegrate the following function $f(x)$.
\n\t
Input the constant of integration as $C$.
Note that \\[\\begin{eqnarray*} &\\int& \\;x^n\\;dx&=&\\frac{x^{n+1}}{n+1}+C,\\;\\;n \\neq -1\\\\ &\\int& \\;\\sin(ax)\\;dx &=& -\\frac{1}{a}\\cos(ax)+C\\\\ &\\int& \\;e^{ax}\\;dx &=& \\frac{1}{a}e^{ax}+C\\\\ \\end{eqnarray*}\\]
\n\tSplitting the integral into three parts and using the above information we have:
\\[\\begin{eqnarray*}\\simplify[std]{Int({b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3},x)}&=&\\simplify[std]{Int({b} * e ^ ({a}*x),x)+Int({b1} * Sin({a1}*x),x)+Int({a2} * x ^ {c3},x) }\\\\ &=&\\simplify[std]{({b}/{a}) * (e ^({a}*x)) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+C} \\end{eqnarray*}\\]
$\\simplify[std]{f(x) = {a5}* x^{an5}+ {b5}*x^{bn5}+ {c5}}$
\n$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]
\nInput all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.
\nClick on Show steps to get more information. You will not lose any marks by doing so.
", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Note that \\[\\begin{eqnarray*} &\\int& \\;x^n\\;dx&=&\\frac{x^{n+1}}{n+1}+C\\\\ \\end{eqnarray*}\\]
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"}, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "$\\simplify[std]{f(x) = {b6} * e ^ ({a6}*x) + {c6}}$
\n$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]
\nInput all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.
\nClick on Show steps to get more information. You will not lose any marks by doing so.
", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Note that \\[\\begin{eqnarray*} &\\int& \\;e^{ax}\\;dx &=& \\frac{1}{a}e^{ax}+C \\end{eqnarray*}\\]
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"}, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "$\\simplify[std]{f(x) = {b7} * Sin({a7}*x) + {b8} * Cos({a8}*x) }$
\n$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]
\nInput all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.
\nClick on Show steps to get more information. You will not lose any marks by doing so.
", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Note that \\[\\begin{eqnarray*} &\\int& \\;\\sin(ax)\\;dx &=& -\\frac{1}{a}\\cos(ax)+C\\\\ &\\int& \\;\\cos(ax)\\;dx &=& \\frac{1}{a}\\sin(ax)+C\\\\ \\end{eqnarray*}\\]
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\n$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]
\nInput all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.
\nClick on Show steps to get more information. You will not lose any marks by doing so.
", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Note that \\[\\begin{eqnarray*} &\\int& \\;\\frac{1}{x}\\;dx&=&\\ln(|x|)+C, \\,\\, {\\rm and} \\,\\, \\frac{1}{x^2}&=x^{-2}& \\end{eqnarray*}\\]
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"}, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "type": "question"}, {"name": "Musa's copy of 3 Integration: solving for constant", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}], "tags": [], "metadata": {"description": "Recovering\noriginal function given some information such as derivative and value at some point.
Find the orginal function $f(x)$ given $f^\\prime (x)$ and value $f(x_0) = C_0;$ that is solve for constant for $\\int f^\\prime (x) \\,dx.$
", "advice": "Note that \\[\\begin{eqnarray*} &\\int& \\;x^n\\;dx&=&\\frac{x^{n+1}}{n+1}+C,\\;\\;n \\neq -1\\\\ &\\int& \\;\\sin(ax)\\;dx &=& -\\frac{1}{a}\\cos(ax)+C\\\\ &\\int& \\;e^{ax}\\;dx &=& \\frac{1}{a}e^{ax}+C\\\\ \\end{eqnarray*}\\]
\n\nFirst integrate:
\n(*) $\\int f^\\prime (x)\\,dx$ = $\\int\\;(a x^n+ c)\\,dx=\\; \\frac{a}{n+1}x^{n+1} + cx +C$
\nthen calculate the value of $C$ from
\n$\\frac{a}{n+1}x_0^{n+1} + cx_0 +C = C_0$
\nand put in back to (*).
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\n$\\displaystyle \\int\\;(\\simplify[std]{ {a5}* x^{an5}+ {c5}})\\,dx=\\;$[[0]]
\n$f (x)=\\;$[[1]]
\nInput all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.
\nClick on Show steps to get more information. You will not lose any marks by doing so.
", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Note that \\[\\begin{eqnarray*} &\\int& \\;x^n\\;dx&=&\\frac{x^{n+1}}{n+1}+C\\\\ \\end{eqnarray*}\\]
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\n$\\displaystyle \\int\\;(\\simplify[std]{ {a} sin({an}x)+ {c}})\\,dx=\\;$[[0]]
\n$f(x)=\\;$[[1]]
\nInput all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.
\nPut 'pi' for $\\pi$ and do not use decimals.
\nClick on Show steps to get more information. You will not lose any marks by doing so.
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