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Integration by parts, trigonometric integrals, partial fractions, complete the square

\n

rebel

\n

rebelmaths

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Integration by Parts

\n

rebelmaths

"}, "tags": ["rebelmaths"], "variable_groups": [], "ungrouped_variables": ["a", "b"], "rulesets": {}, "advice": "

Use Integration by Parts

", "variables": {"b": {"definition": "random(2..4 except a)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "b"}, "a": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "a"}}, "preamble": {"css": "", "js": ""}, "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "

Integration by Parts

", "parts": [{"minValue": "-2", "showCorrectAnswer": true, "prompt": "

Evaluate $\\int_0^\\pi x \\cos(x) \\mathrm{dx}$ using integration by parts, letting $u = x$ and $\\mathrm{dv} = \\cos(x)$

", "correctAnswerFraction": false, "showFeedbackIcon": true, "maxValue": "-2", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "type": "numberentry", "correctAnswerStyle": "plain", "scripts": {}, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false}, {"showCorrectAnswer": true, "gaps": [{"expectedvariablenames": [], "checkingaccuracy": 0.001, "showFeedbackIcon": true, "vsetrange": [0, 1], "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "scripts": {}, "vsetrangepoints": 5, "marks": 1, "answer": "{b}^({a}+1)/({a}+1)", "showpreview": true}, {"expectedvariablenames": [], "checkingaccuracy": 0.001, "showFeedbackIcon": true, "vsetrange": [0, 1], "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "scripts": {}, "vsetrangepoints": 5, "marks": 1, "answer": "-{b}^({a}+1)/({a}+1)^2+1/({a}+1)^2", "showpreview": true}], "variableReplacementStrategy": "originalfirst", "marks": 0, "scripts": {}, "prompt": "

Evaluate $\\int_1^\\var{b}x^\\var{a}\\ln(x)\\mathrm{dx}$ using integration by parts, letting $u = \\ln(x)$ and $\\mathrm{dv} = x^\\var{a}$.

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Answer: [[0]]ln({b})+[[1]]

\n

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Evaluate $\\int_0^{1/2}x\\cos(x)\\mathrm{dx}$ using the substitution $u = x$ and $\\mathrm{dv} = \\cos(\\pi x)\\mathrm{dx}$.

\n

When writing $\\pi$ in your answer simly write pi.

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$\\int_0^{2\\pi}\\sin(\\var{c}t)\\mathrm{dt}$

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$\\int_0^\\var{a} \\frac{y}{e^{\\var{b}y}}\\mathrm{dy}$

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Integration by parts

\n

rebelmaths

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Evaluate the following integrals using integration by parts.

", "preamble": {"js": "", "css": ""}, "advice": "

Use Integration by Parts

", "rulesets": {}, "variable_groups": [], "ungrouped_variables": ["a", "c", "b"], "tags": ["rebelmaths"], "functions": {}, "type": "question"}, {"name": "Partial Fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "TEAME UCC", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/351/"}], "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Partial Fractions

\n

rebelmaths

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "h"], "variables": {"d": {"name": "d", "definition": "random(2..4 except b)", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "h": {"name": "h", "definition": "random(-3..3 except g except -g except 0)", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "c": {"name": "c", "definition": "random(1..5)", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "f": {"name": "f", "definition": "random(1..6 except c)", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "b": {"name": "b", "definition": "random(2..4)", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "g": {"name": "g", "definition": "random(-3..3 except 0)", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "a": {"name": "a", "definition": "random(2..6)", "description": "", "group": "Ungrouped variables", "templateType": "anything"}}, "tags": ["rebelmaths"], "statement": "

Split the following into partial fractions.

", "preamble": {"js": "", "css": ""}, "rulesets": {}, "variable_groups": [], "advice": "

Use Partial Fractions

", "parts": [{"scripts": {}, "gaps": [{"expectedvariablenames": [], "variableReplacements": [], "vsetrangepoints": 5, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "answer": "({a}{c}+{b})/({b}{f}+{c}{d})", "showCorrectAnswer": true, "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "showFeedbackIcon": true, "type": "jme", "scripts": {}, "showpreview": true}, {"expectedvariablenames": [], "variableReplacements": [], "vsetrangepoints": 5, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "answer": "({a}{f}-{d})/({b}{f}+{c}{d})", "showCorrectAnswer": true, "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "showFeedbackIcon": true, "type": "jme", "scripts": {}, "showpreview": true}], "variableReplacementStrategy": "originalfirst", "prompt": "

$\\frac{1+\\var{a}x}{(\\var{b}x-\\var{c})(\\var{d}x+\\var{f})}$

\n

$=$[[0]] $/ (\\var{b}x-\\var{c})+$ [[1]]$/ (\\var{d}x+\\var{f})$

", "marks": 0, "showFeedbackIcon": true, "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true}, {"scripts": {}, "gaps": [{"expectedvariablenames": [], "variableReplacements": [], "vsetrangepoints": 5, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "answer": "{g}/({g}-{h})", "showCorrectAnswer": true, "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "showFeedbackIcon": true, "type": "jme", "scripts": {}, "showpreview": true}, {"expectedvariablenames": [], "variableReplacements": [], "vsetrangepoints": 5, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "answer": "{h}/({h}-{g})", "showCorrectAnswer": true, "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "showFeedbackIcon": true, "type": "jme", "scripts": {}, "showpreview": true}], "variableReplacementStrategy": "originalfirst", "prompt": "

$\\frac{x}{\\simplify{x^2-{g+h}x+{g}{h}}}$

\n

$=$ [[0]]$/(\\simplify{x- {g}}) + $ [[1]]$/(\\simplify{x - {h}})$ 

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Integration using partial fractions

\n

rebelmaths

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$\\int\\frac{1+\\var{a}x}{(x-\\var{c})(x+\\var{f})}\\mathrm{dx}$

\n

$=$[[0]]

\n

Express your answer in terms of the natural log, ln(x). 

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$\\int\\frac{x}{\\simplify{x^2-{g+h}x+{g}{h}}}\\mathrm{dx}$

\n

$=$ [[0]]

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First split into its partial fractions and then integrate

", "statement": "

Split the following into partial fractions and hence evaluate the integrals.

", "type": "question"}, {"name": "Complete the square", "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/"}], "functions": {}, "tags": ["Steps", "algebra", "algebraic manipulation", "complete the square", "completing the square", "quadratics", "steps"], "advice": "\n

Given the quadratic $q(x)=\\simplify{x^2+{2*a}x+ {a^2+b}}$ we complete the square by:

\n

1. Halving the coefficient of $x$ gives $\\var{a}$

\n

2. Work out $\\simplify[all]{p(x)=(x+{a})^2=x^2+{2*a}x+{a^2}}$.
This gives the first two terms of $q(x)$.

\n

3. But the constant term $\\simplify[all]{{a^2}}$ in $p(x)$ is not the same as in $q(x)$, so we need to adjust by adding on $\\simplify[std,!fractionNumbers]{{a^2+b}-{a^2}={b}}$ to $p(x)$.
Hence we get \\[q(x) = \\simplify[all]{p(x)+{b} = (x+{a})^2+{b}}=\\simplify[all]{ (x+{a})^2+{b}}\\]

\n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "

$\\simplify{x^2+{2*a}x+ {a^2+b}} = \\phantom{{}}$ [[0]].

", "gaps": [{"notallowed": {"message": "

Input your answer in the form $(x+a)^2+b$.

", "showstrings": false, "strings": ["x^2", "x*x", "x x", "x(", "x*("], "partialcredit": 0.0}, "checkingaccuracy": 0.0001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 2.0, "answer": "(x+{a})^2+{b}", "type": "jme", "musthave": {"message": "

please input in the form $(x+a)^2+b$

", "showstrings": false, "strings": ["(", ")", "^"], "partialcredit": 0.0}}], "steps": [{"prompt": "\n

Given the quadratic $q(x)=\\simplify{x^2+{2*a}x+ {a^2+b}}$ we complete the square by:

\n

1. Halving the coefficient of $x$ gives $\\var{a}$

\n

2. Work out $\\simplify[all]{p(x)=(x+{a})^2=x^2+{2*a}x+{a^2}}$.
This gives the first two terms of $q(x)$.

\n

3. But the constant term $\\simplify[all]{{a^2}}$ in $p(x)$ is not the same as in $q(x)$ – so we need to adjust by adding on a suitable constant to $p(x)$.

\n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n

Put the following quadratic expression in the form $(x+a)^2+b$ for suitable numbers $a$ and $b$.

\n

Note that you have to input your answer in the form $(x+a)^2+b$  and  the numbers $a,\\;b$ must be input exactly.

\n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "s1*random(1.0..9.5#0.5)", "name": "a"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b": {"definition": "random(1..20)-a^2", "name": "b"}}, "metadata": {"notes": "\n \t\t

5/08/2012:

\n \t\t

Added tags.

\n \t\t

Added description.

\n \t\t

Checked calculation.OK.

\n \t\t", "description": "

Find $c$ and $d$ such that $x^2+ax+b = (x+c)^2+d$.

", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integration and Arctan", "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/"}], "functions": {}, "tags": ["Calculus", "Steps", "arctan", "calculus", "completing the square", "constant of integration", "indefinite integration", "integrals", "integration", "integration by substitution", "standard integrals", "steps", "substitution"], "advice": "\n \n \n

First complete the square for the denominator:
\\[\\begin{eqnarray*} \n \n \\simplify[std]{{a} *x^2 +{2*a*c}*x+{a*c^2+b}}&=&\\simplify[std]{{a}(x^2+{2*c}x+ {a*c^2+b}/{a})}\\\\\n \n &=&\\simplify[std]{{a}((x+{c})^2+ {a*c^2+b}/{a} - {c}^2)}\\\\\n \n &=&\\simplify[std]{{a}((x+{c})^2+ {b}/{a})}\\\\\n \n &=&\\simplify[std]{{a}(x+{c})^2+ {b}}\n \n \\end{eqnarray*}\\]

\n \n \n \n

Make the substitution $\\simplify[std]{y=x+{c}}$ and we get \\[I = \\simplify[std]{Int(1 / ({a} * y ^ 2 + {b}),y)}\\]
Then:
\\[\\begin{eqnarray*} I&=&\\simplify[std]{Int(1 / ({a} * y ^ 2 + {b}),y)}\\\\\n \n &=&\\simplify[std,!simplifyFractions]{1/{a}Int(1 / ( y ^ 2 + ({b}/{a})),y)}\\\\\n \n &=&\\simplify[std]{1/{a}Int(1 / ( y ^ 2 + ({b}/{a})),y)}\\\\\n \n &=&\\simplify[std,!otherNumbers]{1/{a}Int(1 / ( y ^ 2 + {r}^2),y)}\\\\\n \n &=&\\frac{1}{\\var{a}}\\left(\\simplify[std]{((1/{r})*arctan(y/{r}))}\\right)+C\\\\\n \n &=&\\simplify[std]{(1/{r*a})*arctan((x+{c})/{r})+C}\n \n \\end{eqnarray*}\\]
on replacing $y$ by $\\simplify[std]{x+{c}}$.

\n \n \n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n

\\[I=\\simplify[std]{Int(1 / ({a} *x^2 +{2*a*c}*x+{a*c^2+b}),x)}\\]

\n

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

\n

Input all numbers as integers or fractions.

\n

Input the constant of integration as $C$.

\n

You can click on Show steps to get help, you will lose 1 mark if you do so.

\n ", "gaps": [{"notallowed": {"message": "

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

", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 1e-05, "vsetrange": [34.0, 35.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "(1/{r*a})*arctan((x+{c})/{r})+C", "type": "jme"}], "steps": [{"prompt": "

First complete the square in the denominator i.e. write:
\\[\\simplify[std]{{a} *x^2 +{2*a*c}x+{a*c^2+b}}=\\alpha(x+\\beta)^2+\\gamma\\]
for suitable numbers $\\alpha,\\;\\beta,\\;\\gamma$.
Then use the the following standard integral after making a suitable substitution:
\\[\\int \\frac{1}{x^2+a^2}\\;dx=\\frac{1}{a}\\arctan\\left(\\frac{x}{a}\\right)+C\\]

", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n

Find the following integral.

\n

Input the constant of integration as $C$.

\n

Input all numbers as integers or fractions.

\n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..6)", "name": "a"}, "c": {"definition": "s1*random(1..9)", "name": "c"}, "b": {"definition": "r^2*a", "name": "b"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "s": {"definition": "random(2..5)", "name": "s"}, "r": {"definition": "s^2", "name": "r"}}, "metadata": {"notes": "\n \t\t

4/08/2012:

\n \t\t

Added tags.

\n \t\t

Added description.

\n \t\t

Reminded user about entering fractions or integers also about Show steps.

\n \t\t

Changed checking range to 34 to 35 for as otherwise marking too insensitive in evaluation.

\n \t\t

Checked calculation. OK.

\n \t\t", "description": "

Find$\\displaystyle \\int \\frac{dx}{ax^2 +bx+c}$. Results in arctan as $a,\\;b,\\;c$ chosen so that on completing the square $ax^2+bx+c=a(x+r)^2+s,\\;a,\\;s \\gt0$.

", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Definite integration 1", "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/"}], "functions": {}, "tags": ["Calculus", "calculus", "definite integration", "integration", "integration by parts", "integration by parts twice"], "advice": "\n

a)
\\[I=\\int_1^\\var{b1}\\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\\;dx\\]
Expand the parentheses to obtain:

\n

\\[\\begin{eqnarray*}I &=& \\int_1^\\var{b1} \\simplify[std]{{a1 ^ 2} * x ^ 4 + {2 * a1 * c1} * x ^ 3+ {c1 ^ 2+2*a1*d1} * x ^ 2 + {2 * c1 * d1} * x+ {d1 ^ 2} }\\;dx\\\\ &=&\\left[\\simplify[std]{{a1 ^ 2}/5 * x ^ 5 + {2 * a1 * c1}/4 * x ^ 4+ {c1 ^ 2+2*a1*d1}/3 * x ^ 3 + {2 * c1 * d1}/2 * x^2+ {d1 ^ 2}x }\\right]_1^\\var{b1}\\\\ &=&\\var{tans1}\\\\ \\\\&=&\\var{ans1}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]

\n

b)
\\[\\begin{eqnarray*}I&=&\\int_0^{\\var{b2}}\\simplify[std]{1/(x+{m2})}\\;dx\\\\ &=&\\left[\\ln(x+\\var{m2})\\right]_0^{\\var{b2}}\\\\ &=& \\ln(\\var{b2+m2})-\\ln(\\var{m2})\\\\ &=&\\ln\\left(\\frac{\\var{b2+m2}}{\\var{m2}}\\right)\\\\ &=&\\var{ans2}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]

\n

c)
\\[I=\\int_0^\\pi\\simplify[std]{x * ({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\\]
We use integration by parts.

\n

Recall that:
\\[\\int u\\frac{dv}{dx}\\;dx=uv-\\int \\frac{du}{dx}\\;v\\;dx\\]
Here we set $u=x$ and $\\displaystyle \\frac{dv}{dx}=\\simplify[std]{ {w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x)}$

\n

Hence \\[v=\\simplify[std]{({-w}/ {m3}) * Cos({m3} * x) + {1 -w} * (({1-w}/ {m3}) * Sin({m3} * x))}\\]

\n

So \\[\\begin{eqnarray*} I&=&\\left[\\simplify[std]{{-w}*((x / {m3}) * Cos({m3} * x)) + {1 -w} * ((x / {m3}) * Sin({m3} * x))}\\right]_0^\\pi -\\int_0^\\pi\\simplify[std]{ ({ -w} / {m3} )* Cos({m3} * x) + {1 -w} * (1 / {m3} * Sin({m3} * x))}\\;dx\\\\ &=&\\simplify[std]{({-w*cos(m3*pi)})*({pi}/{m3})}-\\left[\\simplify[std]{{ -w} * (1 / {m3 ^ 2})* Sin({m3} * x) -({1 -w} * (1 / {m3 ^ 2}) * Cos({m3} * x))}\\right]_0^\\pi\\\\ &=& \\var{ans3}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]
d)

\n

\\[I=\\int_0^{\\var{b4}}\\simplify[std]{x ^ {m4} * Exp({n4} * x)}\\;dx\\]

\n

Use integration by parts twice with $u=x^2$ and $\\displaystyle \\frac{dv}{dx}=\\simplify[std]{e^({n4}x)}\\Rightarrow v = \\simplify[std]{1/{n4}e^({n4}x)}$
\\[\\begin{eqnarray*} I&=&\\left[\\simplify[std]{x^2/{n4}Exp({n4} * x)}\\right]_0^{\\var{b4}}+\\simplify[std]{2/{abs(n4)}DefInt(x*Exp({n4} * x),x,0,{b4})}\\\\ &=&\\simplify[std]{{b4^2}/{n4}*e^{p}-2/{n4}}\\left(\\left[\\simplify[std]{x/{n4}*e^({n4}*x)}\\right]_0^{\\var{b4}}+\\simplify[std]{1/{abs(n4)}DefInt(e^({n4}x),x,0,{b4})}\\right)\\\\ &=&\\simplify[std]{{b4^2}/{n4}*e^{p}-2/{n4}}\\left(\\simplify[std]{{b4}/{n4}*e^{p}-1/{n4}}\\left[\\simplify[std]{1/{n4}*e^({n4}*x)}\\right]_0^{\\var{b4}}\\right)\\\\ &=&\\simplify[std]{({b4 ^ 2} / {n4}) * Exp({p}) -(({2 * b4} / {n4 ^ 2}) * Exp({p})) + (2 / {n4 ^ 3}) * (Exp({p}) -1)}\\\\ &=&\\var{ans4}\\mbox{ to 4 decimal places} \\end{eqnarray*} \\]

\n ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n

\\[I=\\int_1^{\\var{b1}}\\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\\;dx\\]

\n

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

\n

Input your answer to 2 decimal places.

\n ", "gaps": [{"minvalue": "ans1-tol", "type": "numberentry", "maxvalue": "ans1+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n

\\[I=\\int_0^{\\var{b2}}\\simplify[std]{1/(x+{m2})}\\;dx\\]

\n

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

\n

Input your answer to 2 decimal places.

\n ", "gaps": [{"minvalue": "ans2-tol", "type": "numberentry", "maxvalue": "ans2+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n

\\[I=\\int_0^\\pi\\simplify[std]{x * ({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\\]

\n

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

\n

Input your answer to 2 decimal places.

\n ", "gaps": [{"minvalue": "ans3-tol", "type": "numberentry", "maxvalue": "ans3+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n

\\[I=\\int_0^{\\var{b4}}\\simplify[std]{x ^ {m4} * Exp({n4} * x)}\\;dx\\]

\n

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

\n

Input your answer to 4 decimal places.

\n ", "gaps": [{"minvalue": "ans4-tol1", "type": "numberentry", "maxvalue": "ans4+tol1", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "statement": "

Evaluate the following definite integrals.

", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"ans1": {"definition": "precround(tans1,2)", "name": "ans1"}, "ans2": {"definition": "precround(ln(1+b2/m2),2)", "name": "ans2"}, "ans3": {"definition": "precround(tans3,2)", "name": "ans3"}, "ans4": {"definition": "precround(tans4,4)", "name": "ans4"}, "b4": {"definition": "s7*random(1,2,3)", "name": "b4"}, "b1": {"definition": "random(2..6)", "name": "b1"}, "b2": {"definition": "random(1..20)", "name": "b2"}, "d1": {"definition": "random(-9..9)", "name": "d1"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s7": {"definition": 1.0, "name": "s7"}, "s6": {"definition": -1.0, "name": "s6"}, "m4": {"definition": 2.0, "name": "m4"}, "m3": {"definition": "random(2..9)", "name": "m3"}, "m2": {"definition": "random(1..9)", "name": "m2"}, "tol": {"definition": 0.01, "name": "tol"}, "a1": {"definition": "random(1..7)", "name": "a1"}, "tans4": {"definition": "(e^(p)*(p^2-2*p+2)-2)/(n4^3)", "name": "tans4"}, "c1": {"definition": "t*random(1..9)", "name": "c1"}, "tans1": {"definition": "a1^2*(b1^5-1)/5+a1*c1*(b1^4-1)/2+(2*a1*d1+c1^2)*(b1^3-1)/3+c1*d1*(b1^2-1)+d1^2*(b1-1)", "name": "tans1"}, "tans3": {"definition": "if(w=0,((-1)^(m3)-1)/m3^2,-pi*(-1)^(m3)/m3)", "name": "tans3"}, "tol1": {"definition": 0.0001, "name": "tol1"}, "p": {"definition": "n4*b4", "name": "p"}, "t": {"definition": "random(1,-1)", "name": "t"}, "w": {"definition": "random(0,1)", "name": "w"}, "n4": {"definition": "s6*random(1,2,3)", "name": "n4"}}, "metadata": {"notes": "\n \t\t

3/07/1012:

\n \t\t

Added tags.

\n \t\t

Checked calculations.

\n \t\t

Left tolerances in, as easy to make minor errors in calculations.

\n \t\t

Improved display in Advice.

\n \t\t

Some superscripts e.g. the form a^\\var{b} in latex have to be written as a^{\\var{b}} as not displayed properly (if b has a second digit it slips down). Corrected.

\n \t\t

20/07/2012:

\n \t\t

Set new tolerace variables, tol=0.01, tol1=0.0001.

\n \t\t

Can have expressions in Advice of the form $1\\times E$ where E is an expression. This can be remedied by rewriting - but later as not crucial.

\n \t\t

Added description.

\n \t\t

\n \t\t

25/07/2012:

\n \t\t

 

\n \t\t

Added tags.

\n \t\t

A lot of work in this question - Perhaps it would be more managable broken down into two separate questions?

\n \t\t

 

\n \t\t

Question appears to be working correctly.

\n \t\t

 

\n \t\t

 

\n \t\t", "description": "

Evaluate $\\int_1^{\\,m}(ax ^ 2 + b x + c)^2\\;dx$, $\\int_0^{p}\\frac{1}{x+d}\\;dx,\\;\\int_0^\\pi x \\sin(qx) \\;dx$, $\\int_0^{r}x ^ {2}e^{t x}\\;dx$

", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integration by parts", "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/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "s3", "s2", "s1", "a1", "a2"], "tags": ["Calculus", "Steps", "algebraic manipulation", "calculus", "constant of integration", "integrals", "integrating trigonometric functions", "integration", "integration by parts", "steps"], "preamble": {"css": "", "js": ""}, "advice": "

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

\n

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

\n

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

b)

\n

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

\n

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

\n

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

c)

\n

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

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

", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 1, "prompt": "\n

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

\n

Input all numbers as fractions or integers and not decimals.

\n

Input the constant of integration as $C$.

\n

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

\n ", "marks": 0, "gaps": [{"notallowed": {"message": "

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

", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({a}/{b^2})sin({b}x+{c}) - ({a}/{b})*x*cos({b}x+{c}) + C", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "steps": [{"type": "information", "showCorrectAnswer": true, "scripts": {}, "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 ", "marks": 0}], "type": "gapfill"}, {"prompt": "\n

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

\n

Input all numbers as fractions or integers and not decimals.

\n

Input the constant of integration as $C$.

\n ", "marks": 0, "gaps": [{"notallowed": {"message": "

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

", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({a}/{b})*x*sin({b}x+{c}) + ({a}/{b^2})*cos({b}x+{c}) + C", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n

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

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

\n

Input all numbers as fractions or integers and not decimals.

\n ", "marks": 0, "gaps": [{"notallowed": {"message": "

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

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

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

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

\n ", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "1", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "s3*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "s3": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "a1": {"definition": "s1*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a2": {"definition": "s2*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": ""}}, "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", "description": "

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

", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integration by parts", "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/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "s3", "s2", "s1", "a1", "a2"], "tags": ["Calculus", "Steps", "algebraic manipulation", "calculus", "exponential function", "integrals", "integration", "integration by parts", "integration of exponential function", "steps"], "preamble": {"css": "", "js": ""}, "advice": "

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 $\\displaystyle\\frac{dv}{dx} = \\simplify[std]{e^({c}x)}$.

\n

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

\n

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

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

\n

b)

\n

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

\n

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

\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*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

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

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

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

", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 1, "prompt": "\n

$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

$g(x)=\\;$[[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 ", "marks": 0, "gaps": [{"notallowed": {"message": "

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

", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({a}/{c})*x+{c*b-a}/{c^2}", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "steps": [{"type": "information", "showCorrectAnswer": true, "scripts": {}, "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 ", "marks": 0}], "type": "gapfill"}, {"prompt": "\n

Use the result from the first part to find:

\n

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

\n

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

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

\n

Input all numbers as fractions or integers and not decimals.

\n ", "marks": 0, "gaps": [{"notallowed": {"message": "

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

", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "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}", "marks": 3, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\n

Find the following indefinite integrals.

\n

Input all numbers as fractions or integers and not decimals.

\n ", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "s3*random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "s1*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "s3": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "a1": {"definition": "s1*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a2": {"definition": "s2*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": ""}}, "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", "description": "

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

", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["a", "s1", "b"], "tags": ["arcsin", "Calculus", "calculus", "constant of integration", "integrals", "integration", "integration by substitution", "inverse trigonometric functions", "rebel", "REBEL", "Rebel", "rebelmaths", "standard integrals", "Steps", "steps", "substitution"], "advice": "

Split the integral into two parts
\\[I=\\int\\frac{\\simplify[std]{{a}*x}}{(1-x^2)^{1/2}} \\;dx+\\int\\frac{\\simplify[std]{{b}}}{(1-x^2)^{1/2}} \\;dx\\]  
For the integral \\[I_1=\\int\\frac{\\simplify[std]{{a}*x}}{(1-x^2)^{1/2}} \\;dx \\] use the substitution $u=1-x^2$ and then $du=-2xdx$ and we get
\\[\\begin{eqnarray*}I_1&=&\\simplify[std]{{-a}/2*Int((1 / (u^(1/2))),u)}\\\\\\\\ &=&\\simplify[std]{({-a}/2)*(2u^(1/2))+C}\\\\ &=&\\simplify[std]{({-a})*(1-x^2)^(1/2)+C} \\end{eqnarray*}\\]
The other integral is a standard result: \\[I_2=\\simplify[std]{Int((({b}) / (1-x^2)^(1/2)),x)={b}*arcsin(x)+C}\\]
Putting these together gives:
\\[I=I_1+I_2=\\simplify[std]{-{a}*(1-x^2)^(1/2)+{b}*arcsin(x)+C}\\]

", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 1, "prompt": "\n\t\t\t

\\[I=\\int\\frac{\\simplify[std]{{a}*x+{b}}}{(1-x^2)^{1/2}} \\;dx\\]

\n\t\t\t

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

\n\t\t\t

Input the constant of integration as $C$.

\n\t\t\t

Input all numbers as integers or fractions not as decimals.

\n\t\t\t

 

\n\t\t\t

Click on Show steps if you need help. You will lose 1 mark if you do so.

\n\t\t\t

 

\n\t\t\t \n\t\t\t", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "

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

", "showStrings": false, "strings": ["."], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": [], "checkingaccuracy": 0.0001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "-{a}*(1-x^2)^(1/2)+{b}*arcsin(x)+C", "marks": 3, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 0.9]}], "steps": [{"prompt": "\n\t\t\t\t\t

Split the integral into two parts:
\\[I=\\int\\frac{\\simplify[std]{{a}*x}}{(1-x^2)^{1/2}} \\;dx+\\int\\frac{\\simplify[std]{{b}}}{(1-x^2)^{1/2}} \\;dx\\] 

\n\t\t\t\t\t

Try the substitution $u=1-x^2$ for the first integral and the second one is a standard integral i.e. \\[\\int \\frac{dx}{(1-x^2)^{1/2}}=\\arcsin(x)+C\\]

\n\t\t\t\t\t \n\t\t\t\t\t", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "\n\t

Find the following integral.

\n\t

Input the constant of integration as $C$.

\n\t

Input all numbers as integers or fractions not as decimals.

\n\t \n\t", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "2*s1*random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "b": {"definition": "2*random(1..5)+random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}}, "metadata": {"description": "

Find $\\displaystyle \\int\\frac{ax+b}{(1-x^2)^{1/2}} \\;dx$. Solution involves inverse trigonometric functions.

\n

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Trigonometric Integral", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "TEAME UCC", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/351/"}], "advice": "

Use Integration by Substitution

", "parts": [{"checkingtype": "absdiff", "vsetrange": [0, 1], "type": "jme", "showCorrectAnswer": false, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "answer": "cos(x)^5/5-cos(x)^3/3", "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "variableReplacements": [], "marks": 1, "scripts": {}, "checkvariablenames": false, "prompt": "

Use integration by substitution to evaluate:

\n

$\\int \\sin^3(x)\\cos^2(x)\\mathrm{dx}$

", "expectedvariablenames": []}], "variablesTest": {"condition": "", "maxRuns": 100}, "tags": ["rebelmaths"], "metadata": {"description": "

Trigonometric Integral

\n

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "rulesets": {}, "variables": {}, "statement": "

Trigonometric Integration

", "functions": {}, "ungrouped_variables": [], "preamble": {"css": "", "js": ""}, "type": "question"}, {"name": "Julie's Integration 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["c", "b", "b2", "ans1", "ans2", "t", "tol", "ans", "b1"], "tags": ["areas", "definite integration", "integration", "rebelmaths"], "advice": "\n

First we observe that:\\[\\simplify[std]{int ({t}*exp({b}/{c}*x)+{1-t}*ln({b}x+{c}),x)={t*c}/{b}*exp({b}/{c}*x)+({1-t}/{b}*({b}x+{c})*(ln({b}x+{c})-1))+C}.\\]

\n

Hence we have:

\n

\\[\\begin{eqnarray*} \\simplify[std]{defint ({t}*exp({b}/{c}*x)+{1-t}*ln({b}x+{c}),x,{b1},{b2})}&=&\\left[\\simplify[std]{{t*c}/{b}*exp({b}/{c}*x)+({1-t}/{b}*({b}x+{c})*(ln({b}x+{c})-1))}\\right]_{\\var{b1}}^{\\var{b2}}\\\\&=&\\var{ans}\\end{eqnarray*}\\]

\n

to 3 decimal places.

\n ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "

Enter the area $A$ here to 3 decimal places:

", "allowFractions": false, "variableReplacements": [], "maxValue": "ans+tol", "minValue": "ans-tol", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": "3", "type": "numberentry", "showPrecisionHint": false}], "statement": "

Find the area $A$ of the shape bounded by the $x$-axis, the function $y=\\simplify[std]{{t}*exp({b}/{c}*x)+{1-t}*ln({b}x+{c})}$ and the lines $x=\\var{b1},\\;x=\\var{b2}$.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"c": {"definition": "random(3..7 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..4 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "b2": {"definition": "b1+random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "b2", "description": ""}, "ans1": {"definition": "precround(c*(exp(b*b2/c)-exp(b*b1/c))/b,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans1", "description": ""}, "ans2": {"definition": "precround(1/b*((b*b2+c)*(ln(b*b2+c)-1)-(b*b1+c)*(ln(b*b1+c)-1)),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans2", "description": ""}, "t": {"definition": "random(0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "tol": {"definition": "0.001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "ans": {"definition": "t*ans1+(1-t)*ans2", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}, "b1": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}}, "metadata": {"description": "

Finding areas under graphs using definite integration.

\n

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Julie's copy of Volume of revolution 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "tv", "v", "sb", "sa"], "tags": ["Calculus", "calculus", "definite integration", "diagram", "integral", "integration", "rebelmaths", "rotation about an axis", "rotation about x axis", "volume integral", "volume of revolution"], "advice": "\n

Recall that if $V$ is the volume generated between the limits $x=a$ and $x=b$ by rotating the function about the $x$-axis then $\\displaystyle V=\\pi\\int_a^by^2\\;dx$.

\n

So we have:
\\[\\begin{eqnarray*} V&=&\\pi\\int_{\\var{c}\\pi}^{\\var{c+2}\\pi}\\simplify[std]{{a^2}(cos(x)+{b})^2}\\;dx\\\\ &=&\\var{a^2}\\pi\\int_{\\var{c}\\pi}^{\\var{c+2}\\pi}\\simplify[std]{cos(x)^2+{2*b}*cos(x)+{b^2}}\\;dx\\\\ &=&\\var{a^2}\\pi\\left[\\simplify[std]{(1 / 4) Sin(2*x) + (1 / 2) * x + {2 * b} * Sin(x) + {b ^ 2} * x}\\right]_{\\var{c}\\pi}^{\\var{c+2}\\pi}\\\\ \\end{eqnarray*}\\]
Here we have used the identity $\\cos(x)^2=\\frac{1}{2}(1+\\cos(2x))$ in order to integrate $\\cos(x)^2$.

\n

Since $\\sin(n\\pi)=0$ for all integers $n$ we see that:
\\[\\begin{eqnarray*} V&=&\\var{a^2}\\pi\\frac{\\var{1+2b^2}}{\\var{2}}\\left(\\var{c+2}\\pi-\\var{c}\\pi\\right)\\\\ &=&\\var{a^2*(1+2b^2)}\\pi^2\\\\ &=&\\var{V}\\mbox{ to 3 decimal places} \\end{eqnarray*} \\]

\n \n", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 0, "prompt": "\n

Find the volume of this object.

\n

$V=\\;\\;$[[0]]

\n

Enter your answer to 3 decimal places.

\n

Click on Show steps for information on volumes of revolution. You will not lose any marks.

\n \n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "V", "minValue": "V", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 3, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"variableReplacements": [], "prompt": "

Recall that if $V$ is the volume generated between the limits $x=a$ and $x=b$  then $\\displaystyle V=\\pi\\int_a^by^2\\;dx$.

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "", "marks": 0, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "

Consider the solid object that is obtained when the function: \\[y=\\simplify[std]{{a}(cos(x)+{b})}\\] is rotated by $2\\pi$ radians about the $x$-axis between the limits $x=\\var{c}\\pi$ and $x=\\var{c+2}\\pi$

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "sa*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "sb*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "tv": {"definition": "pi^2*a^2*(1+2*b^2)", "templateType": "anything", "group": "Ungrouped variables", "name": "tv", "description": ""}, "v": {"definition": "precround(tV,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "v", "description": ""}, "sb": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "sb", "description": ""}, "sa": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "sa", "description": ""}}, "metadata": {"description": "

Rotate $y=a(\\cos(x)+b)$ by $2\\pi$ radians about the $x$-axis between $x=c\\pi$ and $x=(c+2)\\pi$. Find the volume of revolution.

\n

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Deirdre Casey", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/681/"}], "extensions": [], "custom_part_types": [], "resources": []}