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Integration by parts, trigonometric integrals, partial fractions, complete the square
\nrebel
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", "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 $\\mathrm{dv} = x^\\var{a}$.
\nAnswer: [[0]]ln({b})+[[1]]
\n", "showFeedbackIcon": true, "type": "gapfill", "variableReplacements": []}, {"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": "1/2*1/pi-1/(pi)^2", "showpreview": true, "prompt": "Evaluate $\\int_0^{1/2}x\\cos(x)\\mathrm{dx}$ using the substitution $u = x$ and $\\mathrm{dv} = \\cos(\\pi x)\\mathrm{dx}$.
\nWhen writing $\\pi$ in your answer simly write pi.
"}], "type": "question"}, {"name": "Integration by Parts 2", "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/"}], "parts": [{"scripts": {}, "checkingaccuracy": 0.001, "showpreview": true, "checkingtype": "absdiff", "showCorrectAnswer": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "prompt": "$\\int_0^{2\\pi}\\sin(\\var{c}t)\\mathrm{dt}$
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", "marks": 1, "expectedvariablenames": [], "type": "jme", "answer": "-{a}e^(-{a}{b})/{b}-e^(-{a}{b})/{b}^2+1/{b}^2", "vsetrange": [0, 1], "checkvariablenames": false, "showFeedbackIcon": true}, {"variableReplacements": [], "scripts": {}, "marks": 0, "type": "information", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"description": "", "group": "Ungrouped variables", "definition": "random(1..3)", "templateType": "anything", "name": "a"}, "c": {"description": "", "group": "Ungrouped variables", "definition": "random(1..8)", "templateType": "anything", "name": "c"}, "b": {"description": "", "group": "Ungrouped variables", "definition": "random(2..6)", "templateType": "anything", "name": "b"}}, "metadata": {"description": "Integration by parts
\nrebelmaths
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Evaluate the following integrals using integration by parts.
", "preamble": {"js": "", "css": ""}, "advice": "Use Integration by Parts
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"}, "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]]$/ (\\var{d}x+\\var{f})$
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\n$=$ [[0]]$/(\\simplify{x- {g}}) + $ [[1]]$/(\\simplify{x - {h}})$
", "marks": 0, "showFeedbackIcon": true, "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true}], "type": "question"}, {"name": "Integration Using 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/"}], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "preamble": {"css": "", "js": ""}, "metadata": {"description": "Integration using partial fractions
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\n$=$[[0]]
\nExpress your answer in terms of the natural log, ln(x).
", "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": 0, "showCorrectAnswer": true, "type": "gapfill", "scripts": {}, "gaps": [{"checkingaccuracy": 0.001, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showpreview": true, "checkvariablenames": false, "showCorrectAnswer": true, "answer": "({a}{c}+1)/({f}+{c})ln(x-{c})+({a}{f}-1)/({c}+{f})ln(x+{f})", "scripts": {}, "vsetrange": [0, 1], "vsetrangepoints": 5, "marks": 1, "checkingtype": "absdiff", "type": "jme", "expectedvariablenames": [], "showFeedbackIcon": true}]}, {"prompt": "$\\int\\frac{x}{\\simplify{x^2-{g+h}x+{g}{h}}}\\mathrm{dx}$
\n$=$ [[0]]
", "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": 0, "showCorrectAnswer": true, "type": "gapfill", "scripts": {}, "gaps": [{"checkingaccuracy": 0.001, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showpreview": true, "checkvariablenames": false, "showCorrectAnswer": true, "answer": "{g}/({g}-{h})ln(x-{g})+{h}/({h}-{g})ln(x-{h})", "scripts": {}, "vsetrange": [0, 1], "vsetrangepoints": 5, "marks": 1, "checkingtype": "absdiff", "type": "jme", "expectedvariablenames": [], "showFeedbackIcon": true}]}], "variable_groups": [], "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "h"], "advice": "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": "\nGiven the quadratic $q(x)=\\simplify{x^2+{2*a}x+ {a^2+b}}$ we complete the square by:
\n1. Halving the coefficient of $x$ gives $\\var{a}$
\n2. Work out $\\simplify[all]{p(x)=(x+{a})^2=x^2+{2*a}x+{a^2}}$.
This gives the first two terms of $q(x)$.
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}}\\]
$\\simplify{x^2+{2*a}x+ {a^2+b}} = \\phantom{{}}$ [[0]].
", "gaps": [{"notallowed": {"message": "Input your answer in the form $(x+a)^2+b$.
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\n1. Halving the coefficient of $x$ gives $\\var{a}$
\n2. Work out $\\simplify[all]{p(x)=(x+{a})^2=x^2+{2*a}x+{a^2}}$.
This gives the first two terms of $q(x)$.
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": "\nPut the following quadratic expression in the form $(x+a)^2+b$ for suitable numbers $a$ and $b$.
\nNote 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\t5/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tChecked 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 \nFirst 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*}\\]
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 $\\simplify[std]{x+{c}}$.
\\[I=\\simplify[std]{Int(1 / ({a} *x^2 +{2*a*c}*x+{a*c^2+b}),x)}\\]
\n$I=\\;$[[0]]
\nInput all numbers as integers or fractions.
\nInput the constant of integration as $C$.
\nYou 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
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\\[\\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\\]
Find the following integral.
\nInput the constant of integration as $C$.
\nInput 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\t4/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tReminded user about entering fractions or integers also about Show steps.
\n \t\tChanged checking range to 34 to 35 for as otherwise marking too insensitive in evaluation.
\n \t\tChecked 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": "\na)
\\[I=\\int_1^\\var{b1}\\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\\;dx\\]
Expand the parentheses to obtain:
\\[\\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*} \\]
\nb)
\\[\\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*} \\]
c)
\\[I=\\int_0^\\pi\\simplify[std]{x * ({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\\]
We use integration by parts.
Recall that:
\\[\\int u\\frac{dv}{dx}\\;dx=uv-\\int \\frac{du}{dx}\\;v\\;dx\\]
Here we set $u=x$ and $\\displaystyle \\frac{dv}{dx}=\\simplify[std]{ {w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x)}$
Hence \\[v=\\simplify[std]{({-w}/ {m3}) * Cos({m3} * x) + {1 -w} * (({1-w}/ {m3}) * Sin({m3} * x))}\\]
\nSo \\[\\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)
\\[I=\\int_0^{\\var{b4}}\\simplify[std]{x ^ {m4} * Exp({n4} * x)}\\;dx\\]
\nUse 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*} \\]
\\[I=\\int_1^{\\var{b1}}\\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\\;dx\\]
\n$I=\\;\\;$[[0]]
\nInput 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]]
\nInput 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]]
\nInput 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]]
\nInput 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\t3/07/1012:
\n \t\tAdded tags.
\n \t\tChecked calculations.
\n \t\tLeft tolerances in, as easy to make minor errors in calculations.
\n \t\tImproved display in Advice.
\n \t\tSome 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\t20/07/2012:
\n \t\tSet new tolerace variables, tol=0.01, tol1=0.0001.
\n \t\tCan 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\tAdded description.
\n \t\t \n \t\t25/07/2012:
\n \t\t\n \t\t
Added tags.
\n \t\tA 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": "The formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]
We choose $u = \\simplify[std]{{a}x}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{sin({b}x+{c})}$.
\nSo $\\displaystyle \\frac{du}{dx}$ = $\\var{a}$ and $v = \\simplify[std]{(-1/{b})*cos({b}x+{c})}$.
\nHence,
\\[ \\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} \\]
For this part we choose $u = \\simplify[std]{{a}x}$ and $\\frac{dv}{dx} = \\simplify[std]{cos({b}x+{c})}$.
\nSo $\\displaystyle \\frac{du}{dx}$ = $\\var{a}$ and $\\displaystyle v = \\simplify[std]{(1/{b})*sin({b}x+{c})}$.
\nHence,
\\[ \\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} \\]
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
$\\displaystyle \\simplify[std]{f(x) = {a2}/{b}*x+{a1}/{b^2}}$
$\\displaystyle \\simplify[std]{g(x) = {-a1}/{b}*x+{a2}/{b^2}}$
", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 1, "prompt": "\n$\\displaystyle \\int \\simplify[std]{{a}x*sin({b}x+{c})} dx = \\phantom{{}}$[[0]]
\nInput all numbers as fractions or integers and not decimals.
\nInput the constant of integration as $C$.
\nYou 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 \nThe formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]
$\\displaystyle \\int \\simplify[std]{{a}x*cos({b}x+{c})} dx = \\phantom{{}}$[[0]]
\nInput all numbers as fractions or integers and not decimals.
\nInput 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": "\nUsing 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 $g(x)$.
$f(x)=\\;$[[0]] $\\;\\;\\;\\;\\;g(x)=\\;$[[1]]
\nInput 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": "\nFind the following indefinite integrals.
\nInput all numbers as fractions or integers and not decimals.
\nInput 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\t3/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tCorrected error in second question answer, + changed to -. Also solution to second gap in third part. Advice changed accordingly.
\n \t\tChecked calculations after corrections. OK.
\n \t\tPenalised use of steps in first part, 1 mark. Added message to that effect.
\n \t\tChanged marks to allow for steps penalty.
\n \t\tImproved 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": "The formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]
We choose $u = \\simplify[std]{{a}x+{b}}$ and $\\displaystyle\\frac{dv}{dx} = \\simplify[std]{e^({c}x)}$.
\nSo $\\displaystyle \\frac{du}{dx} = \\var{a}$ and $\\displaystyle v = \\simplify[std]{(1/{c})*e^({c}*x)}$.
\nHence,
\\[ \\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} \\]
Hence $\\displaystyle \\simplify[std]{g(x)=({a}/{c})*x+{c*b-a}/{c^2}}$
\nFor this part we choose $u = \\simplify[std]{({a}x+{b})^2}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{e^({c}x)}$.
\nSo $\\displaystyle \\frac{du}{dx}$ = $\\simplify[std]{{2*a}*({a}*(x)+{b})}$ and $\\displaystyle v = \\simplify[std]{(1/{c})*e^({c}*x)}$.
\nHence,
\\[ \\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*}\\]
But in Part a we have aready worked out $\\displaystyle \\simplify[std]{Int(({a}x+{b})*e^({c}*x),x)=(({a}/{c})*x+({c*b-a}/{c^2}))*e^({c}*x)+C}$
\nSo 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*}\\]
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 $g(x)$.
$g(x)=\\;$[[0]]
\nInput all numbers as fractions or integers and not decimals.
\nYou 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 \nThe formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]
Use the result from the first part to find:
\n$\\displaystyle I=\\int \\simplify[std]{({a}x+{b})^2*e^({c}x)} dx $
\nYou 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]]
\nInput 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": "\nFind the following indefinite integrals.
\nInput 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\t3/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tChecked calculation. OK.
\n \t\tGot rid of redundant instructions about inputting constant of integration.
\n \t\tPenalised use of steps in first part, 1 mark. Added message to that effect in first part.
\n \t\tAdded message about not inputting decimals in appropriate places.
\n \t\tChanged marks reflecting the use of steps and degree of difficulty in second part.
\n \t\tImproved 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}\\]
\\[I=\\int\\frac{\\simplify[std]{{a}*x+{b}}}{(1-x^2)^{1/2}} \\;dx\\]
\n\t\t\t$I=\\;$[[0]]
\n\t\t\tInput the constant of integration as $C$.
\n\t\t\tInput 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\tSplit 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\\]
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\tFind the following integral.
\n\tInput the constant of integration as $C$.
\n\tInput 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.
\nrebelmaths
", "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
\nrebelmaths
", "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": "\nFirst 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}.\\]
\nHence 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*}\\]
\nto 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.
\nrebelmaths
", "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": "\nRecall that if $V$ is the volume generated between the limits $x=a$ and $x=b$ by rotating the function about the $x$-axis then $\\displaystyle V=\\pi\\int_a^by^2\\;dx$.
\nSo 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 $\\cos(x)^2$.
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*} \\]
Find the volume of this object.
\n$V=\\;\\;$[[0]]
\nEnter your answer to 3 decimal places.
\nClick 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.
\nrebelmaths
", "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": []}