Find 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"], "functions": {}, "question_groups": [{"name": "", "pickQuestions": 0, "pickingStrategy": "all-ordered", "questions": []}], "variables": {"c": {"group": "Ungrouped variables", "definition": "random(1..9)", "templateType": "anything", "description": "", "name": "c"}, "b": {"group": "Ungrouped variables", "definition": "if(b1=a,b1+1,b1)", "templateType": "anything", "description": "", "name": "b"}, "a": {"group": "Ungrouped variables", "definition": "random(1..9)", "templateType": "anything", "description": "", "name": "a"}, "b1": {"group": "Ungrouped variables", "definition": "random(2..9)", "templateType": "anything", "description": "", "name": "b1"}}, "name": "Luis's copy of Indefinite integral by substitution", "ungrouped_variables": ["a", "c", "b", "b1"], "variable_groups": [], "type": "question", "preamble": {"css": "", "js": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"description": "
Find $\\displaystyle \\int \\frac{c}{\\sqrt{a-bx^2}}\\;dx$. Solution involves the inverse trigonometric function $\\arcsin$.
", "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\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"}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "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\\]
", "scripts": {}, "showCorrectAnswer": true, "marks": 0}], "gaps": [{"checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"strings": ["."], "message": "Do not input numbers as decimals, only as integers without the decimal point, or fractions or surds (such as sqrt(2) for $\\sqrt{2}$).
", "partialCredit": 0, "showStrings": false}, "marks": 3, "checkingaccuracy": 0.001, "vsetrangepoints": 5, "showpreview": true, "vsetrange": [0, 0.25], "scripts": {}, "answersimplification": "std", "answer": "({c}/sqrt({b}))*arcsin((sqrt({b})/sqrt({a}))*x)+C", "checkingtype": "absdiff"}], "scripts": {}, "type": "gapfill", "showCorrectAnswer": true, "marks": 0}], "rulesets": {"surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}], "std": ["all", "!collectNumbers", "fractionNumbers"]}, "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 $\\displaystyle \\simplify[std]{(sqrt({b})/sqrt({a}))*x}$