This 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 $x$.
\\[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.
\n ", "gaps": [{"notallowed": {"message": "Input all numbers as integers or fractions and not as decimals.
", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C", "type": "jme"}], "steps": [{"prompt": "Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$
", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\nFind the following integral.
\nInput the constant of integration as $C$.
\n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..5)", "name": "a"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b": {"definition": "s1*random(1..9)", "name": "b"}, "m": {"definition": "random(4..9)", "name": "m"}}, "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\tAdded decimal point as forbidden string and included message in prompt about not entering decimals.
\n \t\tGot rid of a redundant ruleset.
\n \t\t\n \t\t
\n \t\t", "description": "
Find $\\displaystyle \\int x(a x ^ 2 + b)^{m}\\;dx$
", "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": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Steps", "calculus", "indefinite integration", "integration", "integration by substitution", "substitution"], "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*}\\]
A 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\\]
\\[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 \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}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "showpreview": true, "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "ln(abs((({a} * (x ^ 2)) + ({b} * x) + {c})))+C", "checkvariablenames": false, "type": "jme"}], "steps": [{"prompt": "Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$
", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\nFind the following integral.
\nInput the constant of integration as $C$.
\nInput all numbers as integers or fractions.
\n\n \n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..5)", "name": "a"}, "c": {"definition": "a*b1^2+c1", "name": "c"}, "b": {"definition": "2*a+b1", "name": "b"}, "f": {"definition": "-a*(1+b1)^2", "name": "f"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b1": {"definition": "s1*random(1..5)", "name": "b1"}, "test": {"definition": "4*a*c-b^2", "name": "test"}, "c1": {"definition": "max(-10,f+1)+random(1..5)", "name": "c1"}}, "metadata": {"notes": "\n \t\t \t\t
2/08/2012:
\n \t\t \t\tAdded tags.
\n \t\t \t\tAdded description.
\n \t\t \t\tChecked calculation. OK.
\n \t\t \t\tAdded information about Show steps in prompt content area.
\n \t\t \t\tAdded decimal point as forbidden string and included message in prompt about not entering decimals.
\n \t\t \t\tGot rid of a redundant ruleset. !noLeadingMinus added to std ruleset.
\n \t\t \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 \t\t\n \t\t \n \t\t", "description": "
Find $\\displaystyle \\int \\frac{2ax + b}{ax ^ 2 + bx + c}\\;dx$
", "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": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Calculus", "Steps", "calculus", "constant of integration", "integrals", "integrating trigonometric functions", "integration", "integration by substitution", "steps", "substitution"], "advice": "\n \n \nThis 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 \n \n \n\\[\\begin{eqnarray*} I&=&\\simplify[std]{Int((1/{-b})u^{m},u)}\\\\\n \n &=&\\simplify[std]{(1/{-b})u^{m+1}/{m+1}+C}\\\\\n \n &=& \\simplify[std]{({a}+{b}*cos(x))^{m+1}/{-b*(m+1)}+C}\n \n \\end{eqnarray*}\\]
\n \n \n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n\\[I=\\simplify[std]{Int( sin(x)*({a} + {b}*cos(x))^{m},x)}\\]
\nInput all numbers as integers or fractions.
\n$I=\\;$[[0]]
\nInput the constant of integration as $C$.
\nClick on Show steps if you need 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": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "({a}+{b}*cos(x))^{m+1}/{-b*(m+1)}+C", "type": "jme"}], "steps": [{"prompt": "Try the substitution $u=\\simplify[std]{{a}+{b}cos(x)}$
", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\nFind the following integral.
\nInput the constant of integration as $C$.
\nInput all numbers as integers or fractions not as decimals.
\n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..9)", "name": "a"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b": {"definition": "s1*random(1..9)", "name": "b"}, "m": {"definition": "random(3..9)", "name": "m"}}, "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", "description": "Find $\\displaystyle \\int \\sin(x)(a+ b\\cos(x))^{m}\\;dx$
", "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": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Calculus", "Steps", "arcsin", "calculus", "constant of integration", "integrals", "integration", "integration by substitution", "inverse trigonometric functions", "standard integrals", "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$I=\\;$[[0]]
\nInput the constant of integration as $C$.
\nInput all numbers as integers or fractions not as decimals.
\n\n
Click on Show steps if you need help. You will lose 1 mark if you do so.
\n\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": 0.0001, "vsetrange": [0.0, 0.9], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "-{a}*(1-x^2)^(1/2)+{b}*arcsin(x)+C", "type": "jme"}], "steps": [{"prompt": "\nSplit 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 ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\nFind the following integral.
\nInput the constant of integration as $C$.
\nInput all numbers as integers or fractions not as decimals.
\n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "2*s1*random(1..5)", "name": "a"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b": {"definition": "2*random(1..5)+random(1,-1)", "name": "b"}}, "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\tChanged from $\\sqrt{1-x^2}$ to $(1-x^2)^{1/2}$ throughout as display was not good.
\n \t\t\n \t\t", "description": "
Find $\\displaystyle \\int\\frac{ax+b}{(1-x^2)^{1/2}} \\;dx$. Solution involves inverse trigonometric functions.
", "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": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "b1"], "tags": ["Calculus", "Steps", "arcsin", "calculus", "constant of integration", "integrals", "integration", "integration by substitution", "inverse trigonometric functions", "standard integrals", "steps", "substitution"], "preamble": {"css": "", "js": ""}, "advice": "\nFor 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=\\simplify[std]{Int(({c} / (sqrt({a}-{b}x^2))),x)}\\]
\n$I=\\;$[[0]]
\nInput all numbers as integers, fractions or surds. No decimal numbers. You input surds, for example $\\sqrt{2}$, by writing sqrt(2).
Input the constant of integration as $C$.
\nYou can get help by clicking on Show steps. You will lose 1 mark if you do so.
", "marks": 0, "gaps": [{"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, "strings": ["."], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 0.25], "showpreview": true, "marks": 3, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({c}/sqrt({b}))*arcsin((sqrt({b})/sqrt({a}))*x)+C", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "steps": [{"showCorrectAnswer": true, "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\\]
", "scripts": {}, "type": "information", "marks": 0}], "type": "gapfill"}], "statement": "\nFind the following integral.
\n\n ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "if(b1=a,b1+1,b1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "b1": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}}, "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", "description": "Find $\\displaystyle \\int \\frac{c}{\\sqrt{a-bx^2}}\\;dx$. Solution involves the inverse trigonometric function $\\arcsin$.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "showstudentname": true, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "5 questions on using substitution to find indefinite integrals.
"}, "navigation": {"showfrontpage": false, "allowregen": true, "reverse": true, "onleave": {"action": "none", "message": ""}, "preventleave": false, "browse": true, "showresultspage": "oncompletion"}, "showQuestionGroupNames": false, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "duration": 0, "feedback": {"showanswerstate": true, "allowrevealanswer": true, "showactualmark": true, "advicethreshold": 0, "intro": "", "feedbackmessages": [], "showtotalmark": true, "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "type": "exam", "contributors": [{"name": "Gemma Crowe", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2440/"}, {"name": "Jack Dunham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2484/"}], "extensions": [], "custom_part_types": [], "resources": []}