// Numbas version: exam_results_page_options {"questions": [], "duration": 0, "name": "Integration by parts", "showQuestionGroupNames": false, "allQuestions": true, "percentPass": 0, "feedback": {"showanswerstate": true, "advicethreshold": 0, "showactualmark": true, "allowrevealanswer": true, "showtotalmark": true}, "shuffleQuestions": false, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Find indefinite integrals of hyperbolic functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "name": "b", "description": ""}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "name": "b1", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "name": "a", "description": ""}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s2", "description": ""}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "name": "a1", "description": ""}}, "ungrouped_variables": ["a", "b", "s2", "s1", "a1", "b1"], "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

$f(x)=\\simplify[std]{ cosh({a}x+{b})}$

\n

$\\displaystyle{\\int f(x)\\;dx=\\;\\;}$[[0]]

\n

You must include the constant of integration as $C$.

\n

Input all numbers as integers or fractions – not as decimals.

", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"answer": "(1 / {a}) * sinh({a} * x + {b})+C", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "expectedVariableNames": [], "unitTests": [], "notallowed": {"message": "

Input all numbers as integers or fractions – not as decimals.

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "marks": 3, "showFeedbackIcon": true, "scripts": {}, "answerSimplification": "std", "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showPreview": true, "variableReplacements": [], "failureRate": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

$f(x)=\\simplify[std]{x*sinh({a1}x+{b1})}$

\n

$\\displaystyle{\\int f(x)\\;dx=\\;\\;}$[[0]]

\n

Include the constant of integration as $C$.

\n

Input all numbers as integers or fractions – not as decimals.

\n

Please note that if you want to enter a function of the form $xf(x)$ then enter as $x*f(x)$.

", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"answer": "(1 / {a1}) * x * cosh({a1} * x + {b1}) - (1 / ({a1} ^ 2)) * sinh({a1} * x + {b1})+C", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "expectedVariableNames": ["x", "c"], "unitTests": [], "notallowed": {"message": "

Input all numbers as integers or fractions – not as decimals.

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "checkingType": "absdiff", "checkVariableNames": true, "vsetRange": [0, 1], "marks": 3, "showFeedbackIcon": true, "scripts": {}, "answerSimplification": "std", "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showPreview": true, "variableReplacements": [], "failureRate": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\n\n\n

Integrate the following functions $f(x)$.

\n\n\n", "tags": ["Calculus", "calculus", "checked2015", "cosh", "hyperbolic functions", "indefinite integration", "integrating hyperbolic functions", "integration", "integration by parts", "MAS1601", "mas1601", "sinh"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Find  $\\displaystyle \\int\\cosh(ax+b)\\;dx,\\;\\;\\int x\\sinh(cx+d)\\;dx$.

\n

Advice tidied up.

"}, "advice": "

a)  Since $\\int \\cosh(x)=\\sinh(x)+C$ it follows that:
\\[\\int \\simplify[std]{cosh({a}x+{b})}\\;dx = \\frac{1}{\\var{a}}\\simplify[std]{ sinh({a}x+{b})}+C\\]

\n

b)  We perform integration by parts:

\n

Using $\\int u dv = uv - \\int v du$ where:

\n

\\[\\begin{eqnarray*}  &u&=x     \\Rightarrow     du = dx \\\\  &dv& = \\simplify[std]{sinh({a1}x+{b1})}   \\Rightarrow     v= \\simplify[std]{((cosh({a1}x+{b1}))/{a1})}\\end{eqnarray*} \\] 

\n

Hence we have:
\\[\\begin{eqnarray*} \\int \\simplify[std]{x*sinh({a1}x+{b1})}\\;dx&=&\\frac{1}{\\var{a1}}\\simplify[std]{x*cosh({a1}x+{b1})}-\\frac{1}{\\var{a1}} \\int \\simplify[std]{cosh({a1}x+{b1})}\\;dx\\\\ &=&\\frac{1}{\\var{a1}}\\simplify[std]{x*cosh({a1}x+{b1})}-\\frac{1}{\\var{a1^2}}\\simplify[std]{sinh({a1}x+{b1})}+C \\end{eqnarray*} \\]

"}, {"name": "Integration by parts", "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/"}], "variable_groups": [], "variables": {"m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "name": "m"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "name": "c"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s3"}}, "ungrouped_variables": ["s3", "c", "b", "m"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "{b*(m+1)}*x-{c}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n

$I=\\displaystyle \\int \\simplify[std]{x*({b}x+{c})^{m}} dx $
You are given that \\[I=\\simplify[std]{({b}x+{c})^{m+1}/{b^2*(m+1)*(m+2)}*g(x)+C}\\]
For a polynomial $g(x)$. You have to find $g(x)$.

\n

$g(x)=\\;$[[0]]

\n

You can get help by clicking on Show steps. You will lose 1 mark if you do.

\n ", "steps": [{"type": "information", "prompt": "\n \n \n

The formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]
Also you need to know that for $n \\neq -1$:
\\[ \\int (ax+b)^n dx = \\frac{1}{a(n+1)}(ax+b)^{n+1}+C\\]

\n \n \n ", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "

Find the following indefinite integral.

", "tags": ["Calculus", "MAS1601", "Steps", "algebraic manipulation", "checked2015", "indefinite integration", "integration", "integration by parts"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t

3/08/2012:

\n \t\t

Added tags.

\n \t\t

Added description.

\n \t\t

Checked calculation. OK.

\n \t\t

Got rid of redundant instructions about inputting constant of integration.

\n \t\t

Got rid of instruction re not inputting decimals - no restriction needed, so no forbidden strings.

\n \t\t

Penalised use of steps, 1 mark. Added message to that effect.

\n \t\t

Improved Advice display.

\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

Given that $\\displaystyle \\int x({ax+b)^{m}} dx=\\frac{1}{A}(ax+b)^{m+1}g(x)+C$ for a given integer $A$ and polynomial $g(x)$, find $g(x)$.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n

Part a

\n

The formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]

\n

We choose $u = x$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{({b}*x+{c})^{m}}$.

\n

So $\\displaystyle \\frac{du}{dx}$ = $1$ and $\\displaystyle v = \\simplify[std]{(1/{(m+1)*b})*({b}*x+{c})^{m+1}}$.

\n

Hence,
\\[ \\begin{eqnarray*} \\displaystyle \\int \\simplify[std]{x*({b}x+{c})^{m}} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{(x/{(m+1)*b})*({b}*x+{c})^{m+1} - (1/{(m+1)*b})*Int (({b}*x+{c})^{m+1}, x)} \\\\ &=& \\simplify[std]{(x/{(m+1)*b})*({b}*x+{c})^{m+1} - (1/{(m+1)*(m+2)*b^2})*({b}*x+{c})^{m+2}+C} \\\\ &=&\\simplify[std]{({b}*x+{c})^{m+1}/{(m+1)*(m+2)*b^2}*({b*(m+2)}x - ({b}x+{c}))+C}\\\\ &=&\\simplify[std]{({b}*x+{c})^{m+1}/{(m+1)*(m+2)*b^2}*({b*(m+1)}x - {c})+C} \\end{eqnarray*}\\]
The solution is: $\\simplify[std]{g(x)={b*(m+1)}*x-{c}}$.

\n "}, {"name": "Integration by parts", "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/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "name": "c"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s3"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "description": "", "name": "a"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "a1"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "name": "a2"}}, "ungrouped_variables": ["a", "c", "b", "s3", "s2", "s1", "a1", "a2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "({a}/{b^2})sin({b}x+{c}) - ({a}/{b})*x*cos({b}x+{c}) + C", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

Do not input numbers as decimals, only as integers without the decimal point, or fractions

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t

$\\displaystyle \\int \\simplify[std]{{a}x*sin({b}x+{c})} dx = \\phantom{{}}$[[0]]

\n\t\t\t

Input all numbers as fractions or integers and not decimals.

\n\t\t\t

Input the constant of integration as $C$.

\n\t\t\t

You can get help by clicking on Show steps. You will lose 1 mark if you do.

\n\t\t\t", "steps": [{"type": "information", "prompt": "\n\t\t\t\t\t \n\t\t\t\t\t \n\t\t\t\t\t

The formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]

\n\t\t\t\t\t \n\t\t\t\t\t \n\t\t\t\t\t", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "({a}/{b})*x*sin({b}x+{c}) + ({a}/{b^2})*cos({b}x+{c}) + C", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

Do not input numbers as decimals, only as integers without the decimal point, or fractions

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t

$\\displaystyle \\int \\simplify[std]{{a}x*cos({b}x+{c})} dx = \\phantom{{}}$[[0]]

\n\t\t\t

Input all numbers as fractions or integers and not decimals.

\n\t\t\t

Input the constant of integration as $C$.

\n\t\t\t", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{a2}/{b}*x+{a1}/{b^2}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

Do not input numbers as decimals, only as integers without the decimal point, or fractions

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}, {"answer": "{-a1}/{b}*x+{a2}/{b^2}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

Do not input numbers as decimals, only as integers without the decimal point, or fractions

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t

Using the first two parts find:
$\\displaystyle I=\\int \\simplify[std]{{a1}x*sin({b}x+{c})+{a2}x*cos({b}x+{c})} dx $
You are given that \\[I=\\simplify[std]{f(x)*sin({b}x+{c})+g(x)*cos({b}x+{c})+C}\\]
where $f(x)$ and $g(x)$ are polynomials of degree 1. You have to find $f(x)$ and $g(x)$.

\n\t\t\t

$f(x)=\\;$[[0]] $\\;\\;\\;\\;\\;g(x)=\\;$[[1]]

\n\t\t\t

Input all numbers as fractions or integers and not decimals.

\n\t\t\t", "showCorrectAnswer": true, "marks": 0}], "statement": "\n\t

Find the following indefinite integrals.

\n\t

Input all numbers as fractions or integers and not decimals.

\n\t

Input the constant of integration as $C$ where needed.

\n\t", "tags": ["Calculus", "MAS1601", "Steps", "algebraic manipulation", "checked2015", "constant of integration", "integrating trigonometric functions", "integration", "integration by parts"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t

3/08/2012:

\n\t\t

Added tags.

\n\t\t

Added description.

\n\t\t

Corrected error in second question answer, + changed to -. Also solution to second gap in third part. Advice changed accordingly.

\n\t\t

Checked calculations after corrections. OK.

\n\t\t

Penalised use of steps in first part, 1 mark. Added message to that effect.

\n\t\t

Changed marks to allow for steps penalty.

\n\t\t

Improved Advice display.

\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

Find $\\displaystyle \\int x\\sin(cx+d)\\;dx,\\;\\;\\int x\\cos(cx+d)\\;dx $ and hence $\\displaystyle \\int ax\\sin(cx+d)+bx\\cos(cx+d)\\;dx$

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t

a)

\n\t

The formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]

\n\t

We choose $u = \\simplify[std]{{a}x}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{sin({b}x+{c})}$.

\n\t

So $\\displaystyle \\frac{du}{dx}$ = $\\var{a}$ and $v = \\simplify[std]{(-1/{b})*cos({b}x+{c})}$.

\n\t

Hence,
\\[ \\begin{eqnarray} \\int \\simplify[std]{{a}x*sin({b}x+{c})} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{(-{a}/{b})*x*cos({b}x+{c})} - \\int \\left( \\simplify[std]{(-{a}/{b})*cos({b}x+{c})}\\right) dx \\\\ &=& \\simplify[std]{(-{a}/{b})*x*cos({b}x+{c}) + ({a}/{b^2})*sin({b}x+{c}) + C} \\end{eqnarray} \\]

\n\t

b)

\n\t

For this part we choose $u = \\simplify[std]{{a}x}$ and $\\frac{dv}{dx} = \\simplify[std]{cos({b}x+{c})}$.

\n\t

So $\\displaystyle \\frac{du}{dx}$ = $\\var{a}$ and $\\displaystyle v = \\simplify[std]{(1/{b})*sin({b}x+{c})}$.

\n\t

Hence,
\\[ \\begin{eqnarray} \\int \\simplify[std]{{a}x*cos({b}x+{c})} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{({a}/{b})*x*sin({b}x+{c})} - \\int \\left( \\simplify[std]{({a}/{b})*sin({b}x+{c})}\\right) dx \\\\ &=& \\simplify[std]{({a}/{b})*x*sin({b}x+{c}) + ({a}/{b^2})*cos({b}x+{c}) + C} \\end{eqnarray} \\]

\n\t

c)

\n\t

Using the results from Parts a and b, we have \\[\\begin{eqnarray*}I &=& \\int \\simplify[std]{{a1}x*sin({b}x+{c})} dx + \\int \\simplify[std]{{a2}x*cos({b}x+{c})} dx\\\\ &=& \\simplify[std]{{a1}*((-{a}/{b})*x*cos({b}x+{c}) + ({a}/{b^2})*sin({b}x+{c}))+{a2}*(({a}/{b})*x*sin({b}x+{c}) +({a}/{b^2})*cos({b}x+{c}))+C}\\\\ &=&\\simplify[std]{(-{a1}/{b})*x*cos({b}x+{c}) + ({a1}/{b^2})*sin({b}x+{c})+({a2}/{b})*x*sin({b}x+{c}) +({a2}/{b^2})*cos({b}x+{c}) + C}\\\\ &=&\\simplify[std]{({a2}/{b}*x+{a1}/{b^2})*sin({b}x+{c})+({-a1}/{b}*x+{a2}/{b^2})*cos({b}x+{c})+C} \\end{eqnarray*}\\]
Hence
$\\displaystyle \\simplify[std]{f(x) = {a2}/{b}*x+{a1}/{b^2}}$

\n\t

$\\displaystyle \\simplify[std]{g(x) = {-a1}/{b}*x+{a2}/{b^2}}$

\n\t"}, {"name": "Integration by parts", "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/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(2..5)", "description": "", "name": "c"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s3"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "a"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "a1"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "name": "a2"}}, "ungrouped_variables": ["a", "c", "b", "s3", "s2", "s1", "a1", "a2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "({a}/{c})*x+{c*b-a}/{c^2}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

Do not input numbers as decimals, only as integers without the decimal point, or fractions

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t

$I=\\displaystyle \\int \\simplify[std]{({a}x+{b})*e^({c}x)} dx $
You are given that the answer is of the form \\[I=g(x)e^{\\var{c}x}+C\\] for a polynomial $g(x)$. You have to find $g(x)$.

\n\t\t\t

$g(x)=\\;$[[0]]

\n\t\t\t

Input all numbers as fractions or integers and not decimals.

\n\t\t\t

You can get help by clicking on Show steps. You will lose 1 mark if you do.

\n\t\t\t", "steps": [{"type": "information", "prompt": "\n\t\t\t\t\t \n\t\t\t\t\t \n\t\t\t\t\t

The formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]

\n\t\t\t\t\t \n\t\t\t\t\t \n\t\t\t\t\t", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{a^2}/{c}*x^2+{2*a*b*c-2*a^2}/{c^2}*x+{b^2*c^2-2*a*b*c+2*a^2}/{c^3}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

Do not input numbers as decimals, only as integers without the decimal point, or fractions

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t

Use the result from the first part to find:

\n\t\t\t

$\\displaystyle I=\\int \\simplify[std]{({a}x+{b})^2*e^({c}x)} dx $

\n\t\t\t

You are given that the answer is of the form \\[I=h(x)e^{\\var{c}x}+C\\] for a polynomial $h(x)$. You have to find $h(x)$.

\n\t\t\t

$h(x)=\\;$[[0]]

\n\t\t\t

Input all numbers as fractions or integers and not decimals.

\n\t\t\t", "showCorrectAnswer": true, "marks": 0}], "statement": "\n\t

Find the following indefinite integrals.

\n\t

Input all numbers as fractions or integers and not decimals.

\n\t", "tags": ["Calculus", "MAS1601", "Steps", "algebraic manipulation", "checked2015", "exponential function", "integration", "integration by parts", "integration of exponential function"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t

3/08/2012:

\n\t\t

Added tags.

\n\t\t

Added description.

\n\t\t

Checked calculation. OK.

\n\t\t

Got rid of redundant instructions about inputting constant of integration.

\n\t\t

Penalised use of steps in first part, 1 mark. Added message to that effect in first part.

\n\t\t

Added message about not inputting decimals in appropriate places.

\n\t\t

Changed marks reflecting the use of steps and degree of difficulty in second part.

\n\t\t

Improved Advice display.

\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

Given $\\displaystyle \\int (ax+b)e^{cx}\\;dx =g(x)e^{cx}+C$, find $g(x)$. Find $h(x)$, $\\displaystyle \\int (ax+b)^2e^{cx}\\;dx =h(x)e^{cx}+C$. 

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t

a)

\n\t

The formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]

\n\t

We choose $u = \\simplify[std]{{a}x+{b}}$ and $\\displaystyle\\frac{dv}{dx} = \\simplify[std]{e^({c}x)}$.

\n\t

So $\\displaystyle \\frac{du}{dx} = \\var{a}$ and $\\displaystyle v = \\simplify[std]{(1/{c})*e^({c}*x)}$.

\n\t

Hence,
\\[ \\begin{eqnarray} \\int \\simplify[std]{({a}*x+{b})*e^({c}*x)} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{({1}/{c})*({a}x+{b})*e^({c}x) - (1/{c})*Int(({a})*e^({c}x),x)} \\\\ &=& \\simplify[std]{(1/{c})*({a}x+{b})*e^({c}x) -({a}/{c^2})*e^({c}x) + C}\\\\ &=& \\simplify[std]{(({a}x+{b})/{c}-{a}/{c^2})*e^({c}*x) + C}\\\\ &=& \\simplify[std]{(({a}/{c})x+{b*c-a}/{c^2})*e^({c}*x) + C} \\end{eqnarray} \\]

\n\t

Hence $\\displaystyle \\simplify[std]{g(x)=({a}/{c})*x+{c*b-a}/{c^2}}$

\n\t

b)

\n\t

For this part we choose $u = \\simplify[std]{({a}x+{b})^2}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{e^({c}x)}$.

\n\t

So $\\displaystyle \\frac{du}{dx}$ = $\\simplify[std]{{2*a}*({a}*(x)+{b})}$ and $\\displaystyle v = \\simplify[std]{(1/{c})*e^({c}*x)}$.

\n\t

Hence,
\\[ \\begin{eqnarray*}I= \\int \\simplify[std]{({a}*x+{b})^2*e^({c}*x)} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{({1}/{c})*({a}x+{b})^2*e^({c}x) - (1/{c})*Int({2*a}*({a}x+{b})*e^({c}x),x)} \\\\ &=& \\simplify[std]{(1/{c})*({a}x+{b})^2*e^({c}x) -({2*a}/{c})*Int(({a}x+{b})*e^({c}x),x)}\\dots (*) \\end{eqnarray*}\\]

\n\t

But in Part a we have aready worked out $\\displaystyle \\simplify[std]{Int(({a}x+{b})*e^({c}*x),x)=(({a}/{c})*x+({c*b-a}/{c^2}))*e^({c}*x)+C}$ 

\n\t

So on substituting this in equation (*) we find:
\\[ \\begin{eqnarray*}I&=& \\simplify[std]{(1/{c})*({a}x+{b})^2*e^({c}x) -({2*a}/{c})*(({a}/{c})*x+({c*b-a}/{c^2}))*e^({c}*x)+C}\\\\ &=& \\simplify[std]{({a^2}/{c}*x^2+{2*a*b*c-2*a^2}/{c^2}*x+{b^2*c^2-2*a*b*c+2*a^2}/{c^3})*e^({c}x) +C} \\end{eqnarray*}\\]

\n\t

Hence $\\displaystyle \\simplify[std]{h(x)={a^2}/{c}*x^2+{2*a*b*c-2*a^2}/{c^2}*x+{b^2*c^2-2*a*b*c+2*a^2}/{c^3}}$

\n\t"}, {"name": "Integration by parts", "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/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "c"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "name": "d"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "c", "b", "d", "s2", "s1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "({a}*x+{b})/{c}*sin({c}*x+{d})+{a}/{c^2}*cos({c}*x+{d})+C", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

Do not input numbers as decimals, only as integers without the decimal point, or fractions

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n

$I=\\displaystyle \\int \\simplify[std]{({a}x+{b})*cos({c}x+{d})} dx $

\n

$I=\\;$[[0]]

\n

Input all numbers as fractions or integers and not decimals.

\n

You can get help by clicking on Show steps. You will lose 1 mark if you do.

\n ", "steps": [{"type": "information", "prompt": "\n \n \n

The formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]

\n \n \n ", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "(-({a}*x+{b})^2/{c})*cos({c}*x+{d})+(({2*a}({a}x+{b}))/{c^2})*sin({c}*x+{d})+({2*a^2}/{c^3})*cos({c}*x+{d})+C", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

Do not input numbers as decimals, only as integers without the decimal point, or fractions

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 4, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n

Use the result from the first part to find:

\n

$\\displaystyle I=\\int \\simplify[std]{({a}x+{b})^2*sin({c}*x+{d})} dx $

\n

$I=\\;$[[0]]

\n

Input all numbers as fractions or integers and not decimals.

\n ", "showCorrectAnswer": true, "marks": 0}], "statement": "\n

Find the following indefinite integrals.

\n

Input all numbers as fractions or integers and not decimals.

\n

Input the constant of integration as $C$.

\n ", "tags": ["Calculus", "MAS1601", "Steps", "checked2015", "constant of integration", "indefinite integration", "integrating", "integrating trigonometric functions", "integration", "integration by parts", "twice"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t

3/08/2012:

\n \t\t

Added tags.

\n \t\t

Added description.

\n \t\t

Got rid of redundant ruleset, added !noLeadingMinus to std ruleset as we need to keep the standard order for integrating by parts.

\n \t\t

Checked calculation. OK.

\n \t\t

Penalised use of steps in first part, 1 mark. Added message to that effect in first part.

\n \t\t

Added message about not inputting decimals in appropriate places.

\n \t\t

Changed marks reflecting the use of steps and degree of difficulty in second part.

\n \t\t

Improved Advice display.

\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

Find $\\displaystyle \\int (ax+b)\\cos(cx+d)\\; dx $ and hence find $\\displaystyle \\int (ax+b)^2\\sin(cx+d)\\; dx $ 

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n

a)

\n

The formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]

\n

We choose $u = \\simplify[std]{({a}x+{b})}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{cos({c}*x+{d})}$.

\n

So $\\displaystyle \\frac{du}{dx} = \\simplify[std]{{a}}$ and $\\displaystyle v = \\simplify[std]{(1/{c})*sin({c}*x+{d})}$.

\n

Hence,
\\[ \\begin{eqnarray} \\int \\simplify[std]{({a}*x+{b})*cos({c}*x+{d})} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{(({a}*x+{b})/{c})*sin({c}*x+{d}) - ({a}/{c})*Int(sin({c}*x+{d}),x)} \\\\ &=& \\simplify[std]{(({a}x+{b})/{c})*sin({c}*x+{d}) +({a}/{c^2})*cos({c}*x+{d}) + C} \\end{eqnarray} \\]

\n

b)

\n

For this part we choose $u = \\simplify[std]{({a}x+{b})^2}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{sin({c}*x+{d})}$.

\n

So $\\displaystyle \\frac{du}{dx}=\\simplify[std]{{2*a}*({a}*(x)+{b})}$ and $\\displaystyle v = \\simplify[std]{-(1/{c})*cos({c}*x+{d})}$.

\n

Hence,
\\[ \\begin{eqnarray*}I= \\int \\simplify[std]{({a}*x+{b})^2*e^({c}*x)} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{({-1}/{c})*({a}x+{b})^2*cos({c}*x+{d}) + (1/{c})*Int({2*a}*({a}x+{b})*cos({c}*x+{d}),x)} \\\\ &=& \\simplify[std]{({-1}/{c})*({a}x+{b})^2*cos({c}*x+{d}) +({2*a}/{c})*Int(({a}x+{b})*cos({c}*x+{d}),x)}\\dots (*) \\end{eqnarray*}\\]

\n

But in Part a we have aready worked out $\\displaystyle \\simplify[std]{Int(({a}x+{b})*cos({c}*x+{d}),x)=(({a}x+{b})/{c})*sin({c}*x+{d}) +({a}/{c^2})*cos({c}*x+{d})}$ 

\n

So on substituting this in equation (*) we find:
\\[ \\begin{eqnarray*}I&=& \\simplify[std]{({-1}/{c})*({a}x+{b})^2*cos({c}*x+{d}) +({2*a}/{c})*((({a}x+{b})/{c})*sin({c}*x+{d}) +({a}/{c^2})*cos({c}*x+{d}))+C}\\\\ &=& \\simplify[std]{-(({a}*x+{b})^2/{c})*cos({c}*x+{d})+(({2*a}({a}x+{b}))/{c^2})*sin({c}*x+{d})+({2*a^2}/{c^3})*cos({c}*x+{d})+C} \\end{eqnarray*}\\]

\n "}, {"name": "Integration By Parts", "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/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "name": "c"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s3"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "a"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "a1"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "name": "a2"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "name": "t"}}, "ungrouped_variables": ["a", "c", "b", "s3", "s2", "s1", "a1", "a2", "t"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "- ({a}/{b})*x*({t}*cos({b}x+{c})-{1-t}*exp({b}x+{c})) +({a}/{b^2})({t}*sin({b}x+{c})-{1-t}*exp({b}x+{c})) + C", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

Do not input numbers as decimals, only as integers without the decimal point, or fractions

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "

$\\displaystyle \\int \\simplify[std]{{a}x*({t}sin({b}x+{c})+{1-t}*exp({b}x+{c}))} dx = \\phantom{{}}$[[0]]

", "steps": [{"type": "information", "prompt": "\n \n \n

The formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]

\n \n \n \n ", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "

Find the following indefinite integral.

\n

Input all numbers as fractions or integers and not decimals.

\n

You must input the constant of integration as $C$.

\n

Note that if you are entering an expression such as $x*\\cos(p)$ for $p$ some expression then you must enter it as x*cos(p).

", "tags": ["Calculus", "calculus", "checked2015", "exponential function", "indefinite integration", "integration", "integration by parts", "integration of trigonometric functions", "mas1601", "MAS1601"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t

20/06/2012:

\n \t\t \t\t

Added tags.

\n \t\t \t\t

Changed xsin to x*sin in Advice.

\n \t\t \t\t

Tidied up display. Checked calculations. Users have to include capital C for the first two questions.

\n \t\t \t\t

4/07/2012:

\n \t\t \t\t

Edited part c to include brackets around the integrand.

\n \t\t \t\t

Added tags.

\n \t\t \t\t

9/07/2012:

\n \t\t \t\t

Added !noLeadingMinus to ruleset so that integrating by parts expressions are in the expected order.

\n \t\t \t\t

Improved display of last two lines of Advice.

\n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

Integrating by parts.

\n

Find $ \\int ax\\sin(bx+c)\\;dx$ or $\\int ax e^{bx+c}\\;dx$ 

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

The formula for integrating by parts is

\n

\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]

\n

We choose $u = \\simplify[std]{{a}x}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{{t}*sin({b}x+{c})+{1-t}*exp({b}x+{c})}$.

\n

So $\\displaystyle \\frac{du}{dx} = \\var{a}$ and $v = \\simplify[std]{({-t}/{b})*cos({b}x+{c})+{1-t}/{b}*exp({b}x+{c})}$.

\n

Hence,
\\[ \\begin{eqnarray} \\int \\simplify[std]{{a}x*({t}sin({b}x+{c})+{1-t}*exp({b}x+{c})) } dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{({a}/{b})*x*(({-t})*cos({b}x+{c})+{1-t}*exp({b}x+{c}))-{a}/{b}int (({-t})*cos({b}x+{c})+{1-t}*exp({b}x+{c}),x)} \\\\ &=&  \\simplify[std]{({a}/{b})*x*(({-t})*cos({b}x+{c})+{1-t}*exp({b}x+{c}))  + ({t*a}/{b^2})*sin({b}x+{c}) -{(1-t)*a}/{b^2}*exp({b}x+{c})+ C} \\end{eqnarray} \\]

\n

 

"}], "name": "", "pickQuestions": 0}], "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "

Integrate the product of two functions by the method of integration by parts.

"}, "type": "exam", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "browse": true, "showresultspage": "oncompletion", "preventleave": true, "allowregen": true, "showfrontpage": true}, "timing": {"timedwarning": {"action": "none", "message": ""}, "timeout": {"action": "none", "message": ""}, "allowPause": true}, "pickQuestions": 0, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "extensions": [], "custom_part_types": [], "resources": []}