Find an integral by choosing a suitable substitution.
", "licence": "Creative Commons Attribution 4.0 International"}, "percentPass": 0, "showQuestionGroupNames": false, "pickQuestions": 0, "question_groups": [{"questions": [{"name": "Calculate definite integral with a hyperbolic substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "description": "", "name": "tol"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "", "name": "b"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "a"}, "valacc": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(1/a)*ln((2*a+(4a^2-1)^(1/2))/(a+(a^2-1)^(1/2)))", "description": "", "name": "valacc"}, "val": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(valacc,2)", "description": "", "name": "val"}}, "ungrouped_variables": ["a", "b", "tol", "val", "valacc"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"displayType": "radiogroup", "choices": ["$\\textrm{arcsinh}(x)+C$", "$\\textrm{arccosh}(x)+C$", "$\\textrm{arctanh}(x)+C$"], "displayColumns": 0, "prompt": "First, specify the answer to $\\displaystyle{\\int \\frac{1}{\\sqrt{x^2-1}}dx}$:
", "distractors": ["", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 1, "showCorrectAnswer": true, "matrix": [0, 1, 0], "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "val+tol", "minValue": "val-tol", "correctAnswerFraction": false, "marks": 6, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Use your answer from above to evaluate the required integral.
\n$I=\\;\\;$[[0]]
\nInput your answer to $2$ decimal places.
", "showCorrectAnswer": true, "marks": 0}], "statement": "In this question the aim is to use hyperbolic functions to find the value of:
\\[I=\\int_{\\var{b}}^{\\var{2*b}} \\left(\\frac{1}{\\sqrt{\\simplify[std]{{a^2}x^2-{b^2}}}}\\right)\\;dx\\]
30/06/2012:
\nAdded, edited tags
\nSlight change to prompt.
\nCould include standard integral in Show steps (once Show steps is available)
\n19/07/2012:
\nAdded description.
\nChanged Advice on the standard integral - so that it makes sense!
\nAdded Show steps information on the standard integral.
\nChecked calculation.
\nSet new tolerance variable tol=0 for the numeric input.
\n23/07/2012:
\nAdded tags.
\nSolution always requires arccosh(x) and not arcsinh(x) or arctanh(x). Is this on purpose?
\nQuestion appears to be working correctly.
\n22/12/2012:(WHF)
\nChecked calculation, OK. Added tested1 tag.
\nChecked rounding, OK. Added cr1 tag.
\n\n
", "licence": "Creative Commons Attribution 4.0 International", "description": "
Find (hyperbolic substitution):
$\\displaystyle \\int_{b}^{2b} \\left(\\frac{1}{\\sqrt{a^2x^2-b^2}}\\right)\\;dx$
\\[I=\\int_{\\var{b}}^{\\var{2*b}} \\left(\\frac{1}{\\sqrt{\\simplify[std]{{a^2}x^2-{b^2}}}}\\right)\\;dx\\]
\nTo solve this, we will employ the standard integral:\\[\\int \\frac{1}{\\sqrt{x^2-1}}\\;dx=\\simplify{arccosh(x)+C}\\]
\nWe wish to rewrite the integrand of $I$ such that it is of the form $\\displaystyle{\\frac{A}{\\sqrt{x^2-1}}}$ ($A$ is a constant) and hence we can use the standard integral. Let us rearrange the integrand into this form:
\n\\[\\begin{eqnarray*}\\frac{1}{\\sqrt{\\simplify[std]{{a^2}x^2-{b^2}}}} &=& \\frac{1}{\\sqrt{\\simplify[std]{({a}x)^2-{b^2}}}} = \\frac{1}{\\sqrt{\\simplify[std]{{b^2}(({a}x/{b})^2-1)}}} \\\\ &=& \\frac{1}{\\var{b}\\sqrt{\\simplify[std]{(({a}x/{b})^2-1)}}} = \\frac{1}{\\var{b}\\sqrt{\\simplify[std]{(v^2-1)}}}\\end{eqnarray*} \\]
\nwhere we have made the substitution $\\displaystyle{v = \\simplify[std]{{a}x/{b}}}$. Following this then $\\displaystyle{dx = \\simplify[std]{{b}/{a}}}dv$. We can then evaluate our integral in terms of $v$ (being careful to modify their limits of integration from $x$ to $v$):
\n\\[\\begin{eqnarray*} I &=& \\int_{\\var{a}}^{\\var{2*a}}\\frac{1}{\\var{b}\\sqrt{v^2-1}}\\simplify[std]{{b}/{a}}dv \\\\&=&\\frac{1}{\\var{a}}\\int_{\\var{a}}^{\\var{2*a}}\\frac{1}{\\sqrt{v^2-1}}\\;dv\\\\ &=&\\frac{1}{\\var{a}}\\left[\\simplify{arccosh(v)}\\right]_{\\var{a}}^{\\var{2*a}}\\\\ &=&\\var{val} \\end{eqnarray*} \\] to 2 decimal places.
"}, {"name": "Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "b"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..9)", "description": "", "name": "m"}}, "ungrouped_variables": ["a", "s1", "b", "m"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "({a}+{b}*cos(x))^{m+1}/{-b*(m+1)}+C", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Do not input numbers as decimals, only as integers without the decimal point, or fractions.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t\\[I=\\simplify[std]{Int( sin(x)*({a} + {b}*cos(x))^{m},x)}\\]
\n\t\t\tInput all numbers as integers or fractions.
\n\t\t\t$I=\\;$[[0]]
\n\t\t\tInput the constant of integration as $C$.
\n\t\t\tClick on Show steps if you need help. You will lose 1 mark if you do so.
\n\t\t\t", "steps": [{"type": "information", "prompt": "Try the substitution $u=\\simplify[std]{{a}+{b}cos(x)}$
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "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", "tags": ["Calculus", "MAS1601", "Steps", "checked2015", "constant of integration", "integrating trigonometric functions", "integration", "integration by substitution", "substitution"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t2/08/2012:
\n\t\tAdded tags.
\n\t\tAdded description.
\n\t\tChecked calculation. OK.
\n\t\tAdded information about Show steps in prompt content area.
\n\t\tGot rid of a redundant ruleset. !noLeadingMinus added to std ruleset.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\int \\sin(x)(a+ b\\cos(x))^{m}\\;dx$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t \n\t \n\tThis exercise is best solved by using substitution.
Note that if we let $u=\\simplify[std]{{a}+{b}cos(x)}$ then $du=\\simplify[std]{({-b}*sin(x))*dx }$
Hence we can replace $\\sin(x)\\;dx$ by $\\frac{1}{\\var{-b}}\\;du$.
Hence the integral becomes:
\n\t \n\t \n\t \n\t\\[\\begin{eqnarray*} I&=&\\simplify[std]{Int((1/{-b})u^{m},u)}\\\\\n\t \n\t &=&\\simplify[std]{(1/{-b})u^{m+1}/{m+1}+C}\\\\\n\t \n\t &=& \\simplify[std]{({a}+{b}*cos(x))^{m+1}/{-b*(m+1)}+C}\n\t \n\t \\end{eqnarray*}\\]
\n\t \n\t \n\t"}, {"name": "Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "b1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b1=a,b1+1,b1)", "description": "", "name": "b"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "c"}}, "ungrouped_variables": ["a", "c", "b", "b1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "({c}/sqrt({b}))*arcsin((sqrt({b})/sqrt({a}))*x)+C", "vsetrange": [0, 0.25], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Do not input numbers as decimals, only as integers without the decimal point, or fractions or surds (such as sqrt(2) for $\\sqrt{2}$).
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t\\[I=\\simplify[std]{Int(({c} / (sqrt({a}-{b}x^2))),x)}\\]
\n\t\t\t$I=\\;$[[0]]
\n\t\t\tInput all numbers as integers, fractions or surds. No decimal numbers. You input surds, for example, $\\sqrt{2}$ by writing sqrt(2).
\n\t\t\tInput the constant of integration as $C$.
\n\t\t\tYou can get help by clicking on Show steps. You will lose 1 mark if you do so.
\n\t\t\t", "steps": [{"type": "information", "prompt": "Try the substitution $\\displaystyle \\simplify[std]{u=(sqrt({b})/sqrt({a}))*x}$ and then consider the standard integral \\[\\int \\frac{dx}{\\sqrt{1-x^2}}=\\arcsin(x)+C\\]
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n\tFind the following integral.
\n\t\n\t", "tags": ["Calculus", "MAS1601", "Steps", "arcsin", "checked2015", "constant of integration", "integration", "integration by substitution", "inverse trigonometric functions", "standard integrals", "substitution"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t
2/08/2012:
\n\t\tAdded tags.
\n\t\tAdded description.
\n\t\tChecked calculation. OK.
\n\t\tAdded information about Show steps in prompt content area.
\n\t\tCorrected error in Show steps, the substitution was the wrong way round.
\n\t\tSimplified the presentation of Advice.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\int \\frac{c}{\\sqrt{a-bx^2}}\\;dx$. Solution involves the inverse trigonometric function $\\arcsin$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\tFor the integral \\[I=\\simplify[std]{Int((({c}) / (sqrt({a}-{b}x^2))),x)}\\] use the substitution $\\displaystyle \\simplify[std]{u=(sqrt({b})/sqrt({a}))*x}$
so that \\[\\simplify[all,!sqrtProduct,fractionNumbers]{sqrt({a}-{b}x^2)=sqrt({a}-{b}*({a}/{b})*u^2)=sqrt({a}-{a}*u^2)=sqrt({a})*sqrt(1-u^2)}\\]
We have $\\displaystyle \\simplify[std]{du=(sqrt({b})/sqrt({a}))dx}$ and we get
\\[\\begin{eqnarray*}I&=&\\simplify[std]{({c}*(sqrt({a})/sqrt({b})))*Int((1 / ( sqrt({a})*sqrt(1-u^2) )),u)}\\\\ &=&\\simplify[std]{({c}/sqrt({b}))*Int((1 / (sqrt(1-u^2))),u)}\\\\ &=&\\simplify[std]{({c}/sqrt({b}))*arcsin(u)+C}\\\\ &=&\\simplify[std]{({c}/sqrt({b}))*arcsin((sqrt({b})/sqrt({a}))*x)+C} \\end{eqnarray*}\\]
on replacing $u$ by $\\displaystyle \\simplify[std]{(sqrt({b})/sqrt({a}))*x}$
$I=\\;\\;$[[0]]
\nInput your answer to $2$ decimal places.
\nShow steps has some information on the standard integral you may need. You will lose no marks in looking at this.
", "steps": [{"type": "information", "prompt": "Use the standard integral: \\[\\int \\frac{1}{\\sqrt{x^2-1}}\\;dx=\\simplify{arccosh(x)+C}\\]
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "Use hyperbolic functions to find the value of:
\\[I=\\int_{\\var{b}}^{\\var{2*b}} \\left(\\frac{1}{\\sqrt{\\simplify[std]{{a^2}x^2-{b^2}}}}\\right)\\;dx\\]
30/06/2012:
\nAdded, edited tags
\nSlight change to prompt.
\nCould include standard integral in Show steps (once Show steps is available)
\n19/07/2012:
\nAdded description.
\nChanged Advice on the standard integral - so that it makes sense!
\nAdded Show steps information on the standard integral.
\nChecked calculation.
\nSet new tolerance variable tol=0 for the numeric input.
\n23/07/2012:
\n \nAdded tags.
\n \nSolution always requires arccosh(x) and not arcsinh(x) or arctanh(x). Is this on purpose?
\n\n
\n
Question appears to be working correctly.
\n\n
", "licence": "Creative Commons Attribution 4.0 International", "description": "
Find (hyperbolic substitution):
$\\displaystyle \\int_{b}^{2b} \\left(\\frac{1}{\\sqrt{a^2x^2-b^2}}\\right)\\;dx$
This is the standard integral we use: \\[\\int \\frac{1}{\\sqrt{x^2-1}}\\;dx=\\simplify{arccosh(x)+C}\\]
\nFor this example if we make the substitution $\\displaystyle{x = \\simplify[std]{{b}v/{a}}}$ in our integral then we get:
\n\\[\\begin{eqnarray*} I &=&\\frac{1}{\\var{a}}\\int_{\\var{a}}^{\\var{2*a}}\\frac{1}{\\sqrt{v^2-1}}\\;dv\\\\ &=&\\frac{1}{\\var{a}}\\left[\\simplify{arccosh(v)}\\right]_{\\var{a}}^{\\var{2*a}}\\\\ &=&\\var{val} \\end{eqnarray*} \\] to 2 decimal places.
"}, {"name": "Integration by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "b"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "name": "a"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(b^2/4a+random(1..5))", "description": "", "name": "c"}}, "ungrouped_variables": ["a", "c", "b", "s1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "ln(abs((({a} * (x ^ 2)) + ({b} * x) + {c})))+C", "musthave": {"message": "Note that \\[\\int \\frac{1}{x}\\;dx=\\ln(|x|)+C\\] and you must input the absolute value of the argument of the natural logarithm. You input the absolute value using abs, for example abs(x)=$\\simplify{abs(x)}$
", "showStrings": false, "partialCredit": 0, "strings": ["abs"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Do not input numbers as decimals, only as integers without the decimal point, or fractions
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\\[I=\\simplify[std]{Int(({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c}),x)}\\]
\n$I=\\;$[[0]]
\nInput all numbers as integers or fractions.
\nDo not forget to include the constant of integration $C$.
\n ", "steps": [{"type": "information", "prompt": "\nTry the substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$
\nNote that \\[\\int \\frac{1}{x}\\;dx=\\ln(|x|)+C\\] and you must input the absolute value of the argument of the natural logarithm. You input the absolute value using abs, for example abs(x)=$\\simplify{abs(x)}$
\n ", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\nFind the following integral.
\nYou must input the constant of integration as $C$.
\nInput all numbers as integers or fractions.
\nYou can click on Show steps to get a hint. You will lose 1 mark if you do so.
\nNote that $\\displaystyle \\int \\frac{1}{x}\\;dx=\\ln(|x|)+C$ and you must include the absolute value in the argument of $\\ln$. You input $|x|$ as abs(x).
\n ", "tags": ["calculus", "Calculus", "checked2015", "indefinite integral", "indefinite integration", "integration", "integration by substitution", "logarithms", "mas1601", "MAS1601", "natural logarithm", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t20/06/2012:
\n \t\tAdded tags.
\n \t\tExtensively changed the variables in order to simplify the question - quadratics generated which only have complex roots to stop the quadratic being zero in any range.
\n \t\tAdded Steps prompt about losing a mark. Also added to Steps re using abs.
\n \t\tAdded required string abs for answer entry.
\n \t\tImproved spacing.
\n \t\t4/07/2012:
In the steps it is explained that the absolute value of the natural logarithm is inputted as ln(abs(x)) however unless steps are revealed it is unclear how to write |x|. Should this information be included in the question somehow?
\n \t\t6/08/2012:
\n \t\tInformation on using ln(abs(x)) included in the statement.
\n \t\t\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "
Find $\\displaystyle I=\\int \\frac{2 a x + b} {a x ^ 2 + b x + c}\\;dx$ by substitution or otherwise.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\nThis exercise is best solved by using substitution.
\nNote that the numerator $\\simplify[std]{{2 * a} * x + {b}}$ of \\[\\simplify[std]{({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c})}\\] is the derivative of the denominator $\\simplify[std]{{a} * x ^ 2 + {b} * x + {c}}$
\nSo if you use as your substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$ you then have $\\simplify[std]{ du = ({2 * a} * x + {b}) * dx}$
\nHence we can replace $\\simplify[std]{ ({2 * a} * x + {b}) * dx}$ by $du$
\nHence the integral becomes:
\n\\[\\begin{eqnarray*} I&=&\\int\\;\\frac{du}{u}\\\\ &=&\\ln(|u|)+C\\\\ &=& \\simplify[std]{ln(abs({a} * (x ^ 2) + ({b} * x) + {c}))+C} \\end{eqnarray*}\\]
A Useful Result
This example can be generalised.
Suppose \\[I = \\int\\; \\frac{f'(x)}{f(x)}\\;dx\\]
\nThe using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:
\\[I = \\int \\frac{du}{u} = \\ln(|u|)+ C = \\ln(|f(x)|)+C\\]
Do not input numbers as decimals, only as integers without the decimal point, or fractions
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n$I=\\displaystyle \\int \\simplify[std]{({b}*x+{c})/(({a}*x+{d})^{n})} dx$
\nYou are given that \\[I=\\simplify[std]{g(x)*({a}x+{d})^{1-n}}+C\\] for a polynomial $g(x)$.
\nYou have to find $g(x)$.
\n$g(x)=\\;$[[0]]
\nRemember to input all numbers as integers or fractions.
\n ", "showCorrectAnswer": true, "marks": 0}], "statement": "\nFind the following indefinite integral.
\nInput all numbers as integers or fractions, not as decimals.
\n\n ", "tags": ["Calculus", "calculus", "checked2015", "indefinite integral", "indefinite integration", "integration", "integration by substitution", "integration of a rational polynomial", "MAS1601", "substitution"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t
3/7/2012:
Added tags
\n \t\t20/06/2012:
\n \t\tAdded tags.
\n \t\tImproved spacing and display.
\n \t\tGot rid of instruction about including constant of integration as not needed.
\n \t\tChecked calculation.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "$\\displaystyle \\int \\frac{bx+c}{(ax+d)^n} dx=g(x)(ax+d)^{1-n}+C$ for a polynomial $g(x)$. Find $g(x)$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\nLet $y = \\simplify[std]{{a}*x+{d}}$.
\nThen $x=\\frac{1}{\\var{a}}\\simplify[std]{(y-{d})}$ and so we have the numerator $\\simplify[std]{{b}*x+{c}}$ becomes in terms of $y$:
\n$\\simplify[std]{{b}*x+{c} = {b}*1/{a}*(y-{d})+{c}= {m}y+{r}}$ and so
\n\\[\\simplify[std]{({b}*x+{c})/(({a}*x+{d})^{n})} = \\simplify[std]{({m}*y+{r})/(y^{n})={m}/y^{n-1}+{r}/y^{n}}\\]
\nNow,
\\[\\int \\simplify[std]{({b}x+{c})/({a}*x+{d})^{n}} dx = \\int \\left(\\simplify[std]{{m}/y^{n-1}+{r}/y^{n}} \\right)\\frac{dx}{dy} dy \\]
Since $\\displaystyle x = \\simplify[std]{(y-{d})/{a}}$ then $\\displaystyle \\frac{dx}{dy} = \\frac{1}{\\var{a}}$.
\nWe can now calculate the desired integral:
\n\\[ \\begin{eqnarray*} \\int \\left(\\simplify[std]{{m}/y^{n-1}+{r}/y^{n}}\\right) \\frac{dx}{dy} dy &=&\\frac{1}{\\var{a}}\\left(\\int \\simplify[std]{{m}/y^{n-1}}\\;dy+\\int \\simplify[std]{{r}/y^{n}}\\;dy \\right)\\\\ &=&\\frac{1}{\\var{a}}\\left(\\simplify[std]{{-m}/({n-2}*y^{n-2})+ {-r}/({n-1}*y^{n-1})}\\right) + C \\\\ &=& \\simplify[std]{(-{m})/({a*(n-2)}*({a}*x+{d})^{n-2})+(-{r})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C}\\\\ &=&\\simplify[std]{1/({a}x+{d})^{n-1}*(({-m}/{a*(n-2)})*({a}x+{d})-{r}/({a*(n-1)}))}\\\\ &=&\\simplify[std]{1/({a}x+{d})^{n-1}*(({-m}/{(n-2)})*x+{-m*d*(n-1)-r*(n-2)}/{(n-2)*(n-1)*a})} \\end{eqnarray*} \\]
Hence \\[g(x)=\\simplify[std]{({-m}/{(n-2)})*x+{-m*d*(n-1)-r*(n-2)}/{(n-2)*(n-1)*a}}\\]
Input all numbers as integers or fractions and not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\\[I=\\simplify[std]{Int( x*({a} * x ^ 2 + {b})^{m},x)}\\]
\n$I=\\;$[[0]]
\nInput numbers in your answer as integers or fractions and not as decimals.
\nClick on Show steps to get further help. You will lose 1 mark if you do so.
", "steps": [{"type": "information", "prompt": "Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "Find the following integral.
\nInput the constant of integration as $C$.
", "tags": ["Calculus", "calculus", "checked2015", "indefinite integration", "integration", "integration by substitution", "MAS1601", "mas1601", "Steps", "steps", "substitution"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "2/08/2012:
\nAdded tags.
\nAdded description.
\nChecked calculation. OK.
\nAdded information about Show steps in prompt content area.
\nAdded decimal point as forbidden string and included message in prompt about not entering decimals.
\nGot rid of a redundant ruleset.
\n\n
", "licence": "Creative Commons Attribution 4.0 International", "description": "
Find $\\displaystyle \\int x(a x ^ 2 + b)^{m}\\;dx$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n \n \nThis exercise is best solved by using substitution.
Note that if we let $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$ then $du=\\simplify[std]{({2*a} * x)*dx }$
Hence we can replace $xdx$ by $\\frac{1}{\\var{2*a}}du$.
Hence the integral becomes:
\n \n \n \n\\[\\begin{eqnarray*} I&=&\\simplify[std]{Int((1/{2*a})u^{m},u)}\\\\\n \n &=&\\simplify[std]{(1/{2*a})u^{m+1}/{m+1}+C}\\\\\n \n &=& \\simplify[std]{({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C}\n \n \\end{eqnarray*}\\]
\n \n \n \nA Useful Result
This example can be generalised.
Suppose \\[I = \\int\\; f'(x)g(f(x))\\;dx\\]
The using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:
\\[I = \\int g(u)\\;du \\]
And if we can find this simpler integral in terms of $u$ we can replace $u$ by $f(x)$ and get the result in terms of $x$.
Do not input numbers as decimals, only as integers without the decimal point, or fractions
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\\[I=\\simplify[std]{Int(({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c}),x)}\\]
\n$I=\\;$[[0]]
\nInput the constant of integration as $C$.
\nInput all numbers as integers or fractions not as decimals.
\nClick on Show steps if you need help. You will lose 1 mark if you do so.
\n ", "steps": [{"type": "information", "prompt": "Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\nFind the following integral.
\nInput the constant of integration as $C$.
\nInput all numbers as integers or fractions.
\n\n ", "tags": ["Calculus", "calculus", "checked2015", "indefinite integration", "integration", "integration by substitution", "MAS1601", "mas1601", "Steps", "steps", "substitution"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t
2/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tChecked calculation. OK.
\n \t\tAdded information about Show steps in prompt content area.
\n \t\tAdded decimal point as forbidden string and included message in prompt about not entering decimals.
\n \t\tGot rid of a redundant ruleset. !noLeadingMinus added to std ruleset.
\n \t\tNote that the choice of variables means that the argument of the log answer is always $\\gt 0$ so no need to use abs.
\n \t\t\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "
Find $\\displaystyle \\int \\frac{2ax + b}{ax ^ 2 + bx + c}\\;dx$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "This exercise is best solved by using substitution.
\nNote that the numerator $\\simplify[std]{{2 * a} * x + {b}}$ of \\[\\simplify[std]{({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c})}\\] is the derivative of the denominator $\\simplify[std]{{a} * x ^ 2 + {b} * x + {c}}$
\nSo if you use as your substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$ you then have $\\simplify[std]{ du = ({2 * a} * x + {b}) * dx}$
\nHence we can replace $\\simplify[std]{ ({2 * a} * x + {b}) * dx}$ by $du$
\nHence the integral becomes:
\n\\[\\begin{eqnarray*} I&=&\\int\\;\\frac{du}{u}\\\\ &=&\\ln(|u|)+C\\\\ &=& \\simplify[std]{ln(abs({a} * (x ^ 2) + ({b} * x) + {c}))+C} \\end{eqnarray*}\\]
\nA Useful Result
This example can be generalised.
Suppose \\[I = \\int\\; \\frac{f'(x)}{f(x)}\\;dx\\]
The using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:
\\[I = \\int \\frac{du}{u} = \\ln(|u|)+ C = \\ln(|f(x)|)+C\\]