// Numbas version: exam_results_page_options
{"name": "Maths Support: Hyperbolic functions", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "preventleave": false, "browse": true, "showfrontpage": false, "showresultspage": "never"}, "duration": 0.0, "metadata": {"notes": "", "description": "<p>4 questions involving hyperbolic functions.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "shufflequestions": false, "questions": [], "percentpass": 50.0, "allQuestions": true, "pickQuestions": 0, "type": "exam", "feedback": {"showtotalmark": true, "advicethreshold": 0.0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Hyperbolic Functions 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": ["cosh", "hyperbolic equations", "hyperbolic functions", "logarithms", "quadratic equation", "sinh", "solving equations", "solving hyperbolic equations", "solving quadratic equation"], "advice": "\n    <p>Using \\[\\cosh(x)=\\frac{e^x+e^{-x}}{2},\\;\\;\\;\\sinh(x)=\\frac{e^x-e^{-x}}{2}\\] and substituting into the equation gives:</p>\n    <p>\\[\\simplify[std]{{a1+b1}e^x+{a1-b1}e^(-x)={2*c1}}\\]</p>\n    <p>On multiplying throughout by $e^x$, putting $y=e^x$ and tidying up the equation, we get:</p>\n    <p>\\[\\simplify[std]{{m1}* y ^ 2 + {m2} * y + {m3} = 0}\\]</p>\n    <p>This quadratic has solutions:</p>\n    <p>\\[y = \\simplify[std]{{d1}/{al1}},\\;\\;\\;y=\\simplify[std]{{d2}/{be1}}\\]</p>\n    <p>Since $y=e^x$ we see that the solutions in terms of $x$ are:<br/>\\[x = \\ln\\left(\\simplify[std]{{d1}/{al1}}\\right)=\\var{tans2},\\;\\;\\;x=\\ln\\left(\\simplify[std]{{d2}/{be1}}\\right)=\\var{tans3}\\]<br/>both to 3 decimal places.</p>\n    <p>Hence the minimum solution is $x=\\var{ans2}$ and the maximum solution is $x=\\var{ans3}$.</p>\n    ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      \n      \n      <p>Input the solutions for $x$ here:(if the solutions are the same input the number twice)</p>\n      \n      \n      \n      <p>Least solution = [[0]]</p>\n      \n      \n      \n      <p>Greatest solution= [[1]]</p>\n      \n      \n      \n      <p>Input both to 3 decimal places.</p>\n      \n      \n        ", "gaps": [{"minvalue": "ans2-tol", "type": "numberentry", "maxvalue": "ans2+tol", "marks": 1.0, "showPrecisionHint": false}, {"minvalue": "ans3", "type": "numberentry", "maxvalue": "ans3", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "statement": "\n    <p>Solve the following equation for $x$.</p>\n    <p>\\[\\simplify[std]{{a1}cosh(x)+{b1}sinh(x)={c1}}\\]</p>\n    ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"ans2": {"definition": "precround(ans12,3)", "name": "ans2"}, "ans3": {"definition": "precround(ans21,3)", "name": "ans3"}, "b1": {"definition": "s*(al1*be1-d1*d2)/2", "name": "b1"}, "d2": {"definition": "s6*random(1..8)", "name": "d2"}, "ans21": {"definition": "max(tans2,tans3)", "name": "ans21"}, "d1": {"definition": "if(al1*be1=d*d2,d+2,d)", "name": "d1"}, "al1": {"definition": "s5*random(2,4,6)", "name": "al1"}, "s6": {"definition": "random(1,-1)", "name": "s6"}, "s5": {"definition": 1.0, "name": "s5"}, "m1": {"definition": "round((a1+b1)/h)", "name": "m1"}, "m3": {"definition": "round((a1-b1)/h)", "name": "m3"}, "m2": {"definition": "round(-2*c1/h)", "name": "m2"}, "be1": {"definition": "s6*random(1..9)", "name": "be1"}, "tol": {"definition": 0.0, "name": "tol"}, "ans12": {"definition": "min(tans2,tans3)", "name": "ans12"}, "a2": {"definition": "(al1*be1+d1*d2)/2", "name": "a2"}, "c1": {"definition": "s*(d1*be1+d2*al1)/2", "name": "c1"}, "tans3": {"definition": "precround(ln(d2/be1),3)", "name": "tans3"}, "tans2": {"definition": "precround(ln(d1/al1),3)", "name": "tans2"}, "a1": {"definition": "abs(a2)", "name": "a1"}, "d": {"definition": "s5*random(2,4)", "name": "d"}, "h": {"definition": "gcf(gcf(abs(a1+b1),abs(a1-b1)),abs(2*c1))", "name": "h"}, "s": {"definition": "sign(a2)", "name": "s"}}, "metadata": {"notes": "\n                \t\t<p><strong>2/07/2012:</strong></p>\n                \t\t    <p>Added tags.</p>\n                \t\t    <p>Forced answers to be exactly to 3 decimal places, no tolerances via tolerance variable tol=0.</p>\n                \t\t    <p>Improved display in Advice. Checked calculations.</p>\n                \t\t    <p><strong>19/07/2012:</strong></p>\n                \t\t    <p>Added description. Rechecked calculations.</p>\n                \t\t    <p>&nbsp;</p>\n                \t\t    <p><strong>25/07/2012:</strong></p>\n                \t\t    <p>&nbsp;Added tags.</p>\n                \t\t    <p>&nbsp;Question appears to be working correctly.</p>\n                \t\t    <p>&nbsp;</p>\n                \t\t", "description": "<p>Solve for $x$: $a\\cosh(x)+b\\sinh(x)=c$. There are two solutions for this example.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Hyperbolic Functions 2", "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", "b", "n", "a1", "a2", "b1", "b2"], "tags": ["Calculus", "Differentiation", "calculus", "chain rule", "cosh", "derivatives of hyperbolic functions", "differential", "differentiate", "differentiating hyperbolic functions", "differentiation", "hyperbolic functions", "product rule", "sinh", "tanh"], "preamble": {"css": "", "js": ""}, "advice": "<p>Here is a table of the derivatives of some of the hyperbolic functions:</p>\n<table border=\"1\">\n  <thead>\n    <tr><th>$f(x)$</th><th>$\\displaystyle{\\frac{df}{dx}}$</th></tr>\n  </thead>\n  <tbody>\n    <tr>\n      <td>$\\sinh(bx)$</td>\n      <td>$b\\cosh(bx)$</td>\n    </tr>\n    <tr>\n      <td>$\\cosh(bx)$</td>\n      <td>$b\\sinh(bx)$</td>\n    </tr>\n    <tr>\n      <td>$\\tanh(bx)$</td>\n      <td>$\\simplify{b*sech(bx)^2}$</td>\n    </tr>\n  </tbody>\n</table>\n<h4>a)</h4>\n<p>$f(x)=\\simplify[std]{ x ^ {n} * sinh({a1} * x + {b1})}$</p>\n<p>Use the product rule to obtain:<br/>\\[\\frac{df}{dx} = \\simplify[std]{{n} * (x ^ {(n -1)}) * sinh({a1} * x + {b1}) + {a1} * (x ^ {n}) * Cosh({a1} * x + {b1})}\\]</p>\n<h4>b)</h4>\n<p>$f(x)=\\tanh(\\simplify[std]{{a}x+{b}})$</p>\n<p>Using the table above we get:<br/>\\[\\frac{df}{dx} = \\simplify[std]{{a}*sech({a}x+{b})^2}\\]</p>\n<h4>c)</h4>\n<p>$f(x)=\\ln(\\cosh(\\simplify[std]{{a2}x+{b2}}))$</p>\n<p>Using the chain rule we find:</p>\n<p>\\[\\frac{df}{dx} = \\simplify[std]{{a2} * tanh({a2} * x + {b2})}\\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n        <p>$f(x)=\\simplify[std]{ x ^ {n} * sinh({a1} * x + {b1})}$</p>\n      <p>$\\displaystyle{\\frac{df}{dx}=\\;\\;}$[[0]]</p>\n        ", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{n} * (x ^ {(n -1)}) * sinh({a1} * x + {b1}) + {a1} * (x ^ {n}) * Cosh({a1} * x + {b1})", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n        <p>$f(x)=\\tanh(\\simplify[std]{{a}x+{b}})$</p>\n      <p>$\\displaystyle{\\frac{df}{dx}=\\;\\;}$[[0]]</p>\n        ", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{a}*sech({a}x+{b})^2", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n        <p>$f(x)=\\ln(\\cosh(\\simplify[std]{{a2}x+{b2}}))$</p>\n      <p>$\\displaystyle{\\frac{df}{dx}=\\;\\;}$[[0]]</p>\n        ", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{a2} * tanh({a2} * x + {b2})", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "<p>Write down the derivatives of the following functions $f(x)$ .</p>\n<p>Note that in order to input the square of a function such as $\\sinh(x)$ you have to input it as <code>sinh(x))^2</code>, similarly for the other hyperbolic functions.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "random(-9..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "n": {"definition": "random(3..7)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "a1": {"definition": "random(-9..-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a2": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": ""}, "b1": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "b2": {"definition": "random(-9..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b2", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>29/06/2012:</strong></p>\n        \t\t    <p>Added and edited tags.</p>\n        \t\t    <p><strong>19/07/2012:</strong></p>\n        \t\t    <p>Added description.</p>\n        \t\t    <p>There is also the problem of inputting functions of the form $xf(x)$ if $n=1$ or $2$ in the first question. So have reset $n$ to between $3$ and $7$. Otherwise would have to have an instruction here (perhaps depending on value of $n$).</p>\n        \t\t    <p>Checked calculation.</p>\n        \t\t    <p><strong>23/07/2012:</strong></p>\n        \t\t    <p>Added tags.</p>\n        \t\t    <p></p>\n        \t\t    <p>Question appears to be working correctly.</p>\n        \t\t    <p>&nbsp;</p>\n        \t\t", "description": "<p>Differentiate the following functions: $\\displaystyle x ^ n \\sinh(ax + b),\\;\\tanh(cx+d),\\;\\ln(\\cosh(px+q))$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Hyperbolic Functions 3", "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", "b", "s2", "s1", "a1", "b1"], "tags": ["Calculus", "calculus", "cosh", "hyperbolic functions", "indefinite integration", "integral", "integrating hyperbolic functions", "integration", "integration by parts", "sinh"], "preamble": {"css": "", "js": ""}, "advice": "<h4>a)</h4>\n<p>\\[\\int \\simplify[std]{cosh({a}x+{b})}\\;dx = \\frac{1}{\\var{a}}\\simplify[std]{ sinh({a}x+{b})}+C\\]</p>\n<h4>b)</h4>\n<p>Integrate by parts, so that</p>\n<p>\\[\\begin{eqnarray*} \\int \\simplify[std]{x*sinh({a1}x+{b1})}\\;dx&amp;=&amp;\\int \\simplify[std]{x*d((cosh({a1}x+{b1}))/{a1})}\\\\ &amp;=&amp;\\frac{1}{\\var{a1}}\\simplify[std]{x*cosh({a1}x+{b1})}-\\frac{1}{\\var{a1}} \\int \\simplify[std]{cosh({a1}x+{b1})}\\;dx\\\\ &amp;=&amp;\\frac{1}{\\var{a1}}\\simplify[std]{x*cosh({a1}x+{b1})}-\\frac{1}{\\var{a1^2}}\\simplify[std]{sinh({a1}x+{b1})}+C \\end{eqnarray*} \\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n        <p>$f(x)=\\simplify[std]{ cosh({a}x+{b})}$</p>\n      <p>$\\displaystyle{\\int f(x)\\;dx=\\;\\;}$[[0]]</p>\n      <p>You must include the constant of integration as $C$.</p>\n      <p>Input all numbers as integers or fractions &ndash; not as decimals.</p>\n        ", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Input all numbers as integers or fractions &ndash; not as decimals.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "(1 / {a}) * sinh({a} * x + {b})+C", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>$f(x)=\\simplify[std]{x*sinh({a1}x+{b1})}$</p>\n<p>$\\displaystyle{\\int f(x)\\;dx=\\;\\;}$[[0]]</p>\n<p>Include the constant of integration as $C$.</p>\n<p>Input all numbers as integers or fractions &ndash; not as decimals.</p>\n<p>Please note that if you want to enter a function of the form $xf(x)$ then enter as <code>x*f(x)</code>&nbsp;so that it's clear what you&nbsp;mean.</p>", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Input all numbers as integers or fractions &ndash; not as decimals.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "(1 / {a1}) * x * Cosh({a1} * x + {b1}) - (1 / ({a1} ^ 2)) * Sinh({a1} * x + {b1})+C", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\n    \n    \n    <p>Integrate the following functions $f(x)$.</p>\n    \n    \n    ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "s1*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "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": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "b1": {"definition": "s2*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>29/06/2012:</strong></p>\n        \t\t    <p><strong><br /></strong>Added and edited tags.</p>\n        \t\t    <p>Added forbidden strings, the decimal point, to ensure fraction or integer input for numbers.</p>\n        \t\t    <p>Put in $C$ as integration constant in both answers.</p>\n        \t\t    <p><strong>19/07/2012:</strong></p>\n        \t\t    <p>Added instruction in second question so that an input of the form $x\\cosh$ is recognized.</p>\n        \t\t    <p>Added description.</p>\n        \t\t    <p>Checked calculation.</p>\n        \t\t    <p><strong>23/07/2012:</strong></p>\n        \t\t    <p></p>\n        \t\t    <p>Added tags.</p>\n        \t\t    <p>&nbsp;</p>\n        \t\t    <p>&nbsp;</p>\n        \t\t    <p>&nbsp;</p>\n        \t\t", "description": "<p>Find &nbsp;$\\displaystyle \\int\\cosh(ax+b)\\;dx,\\;\\;\\int x\\sinh(cx+d)\\;dx$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Hyperbolic Functions 4", "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", "b", "valacc", "val", "tol"], "tags": ["Calculus", "Steps", "arccosh", "calculus", "definite integration", "hyperbolic functions", "integral", "integration", "integration by substitution", "inverse hyperbolic functions", "standard integrals", "steps", "substitution"], "preamble": {"css": "", "js": ""}, "advice": "\n    <p>This is the standard integral we use: \\[\\int \\frac{1}{\\sqrt{x^2-1}}\\;dx=\\simplify{arccosh(x)+C}\\]</p>\n    <p>For this example if we make the substitution $\\displaystyle{x = \\simplify[std]{{b}v/{a}}}$ in our integral then we get:</p>\n    <p>\\[\\begin{eqnarray*} I &amp;=&amp;\\frac{1}{\\var{a}}\\int_{\\var{a}}^{\\var{2*a}}\\frac{1}{\\sqrt{v^2-1}}\\;dv\\\\ &amp;=&amp;\\frac{1}{\\var{a}}\\left[\\simplify{arccosh(v)}\\right]_{\\var{a}}^{\\var{2*a}}\\\\ &amp;=&amp;\\var{val} \\end{eqnarray*} \\] to 2 decimal places.</p>\n    ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 0, "prompt": "\n        <p>$I=\\;\\;$[[0]]</p>\n        <p>Input your answer to $2$ decimal places.</p>\n        <p>Show steps has some information on the standard integral you may need. You will lose no marks in looking at this.</p>\n        ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 3, "maxValue": "val+tol", "minValue": "val-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "steps": [{"showCorrectAnswer": true, "prompt": "<p>Use the standard integral:</p>\n<p>\\[\\int \\frac{1}{\\sqrt{x^2-1}}\\;dx=\\simplify{arccosh(x)+C}\\]</p>", "scripts": {}, "type": "information", "marks": 0}], "type": "gapfill"}], "statement": "<p>Use hyperbolic functions to find the value of: <br/>\\[I=\\int_{\\var{b}}^{\\var{2*b}} \\left(\\frac{1}{\\sqrt{\\simplify[std]{{a^2}x^2-{b^2}}}}\\right)\\;dx\\]</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "random(1..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "val": {"definition": "precround(valacc,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "val", "description": ""}, "valacc": {"definition": "(1/a)*ln((2*a+(4a^2-1)^(1/2))/(a+(a^2-1)^(1/2)))", "templateType": "anything", "group": "Ungrouped variables", "name": "valacc", "description": ""}, "tol": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>30/06/2012:</strong></p>\n        \t\t        <p>Added, edited tags</p>\n        \t\t        <p>Slight change to prompt.</p>\n        \t\t        <p>Could include standard integral in Show steps (once Show steps is available)</p>\n        \t\t        <p><strong>19/07/2012:</strong></p>\n        \t\t        <p>Added description.</p>\n        \t\t        <p>Changed Advice on the standard integral - so that it makes sense!</p>\n        \t\t        <p>Added Show steps information on the standard integral.</p>\n        \t\t        <p>Checked calculation.</p>\n        \t\t        <p>Set new tolerance variable tol=0 for the numeric input.</p>\n        \t\t        <p><strong>23/07/2012:</strong></p>\n        \t\t        <p></p>\n        \t\t        <p>Added tags.</p>\n        \t\t        <p></p>\n        \t\t        <p>Solution always requires arccosh(x) and not arcsinh(x) or arctanh(x). Is this on purpose?</p>\n        \t\t        <p>&nbsp;</p>\n        \t\t        <p>&nbsp;</p>\n        \t\t        <p>Question appears to be working correctly.</p>\n        \t\t        <p>&nbsp;</p>\n        \t\t        <p><br /><br /></p>\n        \t\t        <p>&nbsp;</p>\n        \t\t", "description": "<p>Find (hyperbolic substitution):<br />$\\displaystyle \\int_{b}^{2b} \\left(\\frac{1}{\\sqrt{a^2x^2-b^2}}\\right)\\;dx$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": [], "custom_part_types": [], "resources": []}