// Numbas version: exam_results_page_options
{"name": "Integral by hyperbolic substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "description": "", "name": "tol"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..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", "valacc", "val", "tol"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Integral by hyperbolic substitution", "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "val+tol", "minValue": "val-tol", "correctAnswerFraction": false, "marks": 3, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<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>", "steps": [{"type": "information", "prompt": "<p>Use the standard integral: \\[\\int \\frac{1}{\\sqrt{x^2-1}}\\;dx=\\simplify{arccosh(x)+C}\\]</p>", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "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>", "tags": ["arccosh", "Calculus", "checked2015", "definite integration", "hyperbolic functions", "integration", "integration by substitution", "inverse hyperbolic functions", "MAS1601", "standard integrals", "Steps", "substitution"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>30/06/2012:</strong></p>\n        <p>Added, edited tags</p>\n        <p>Slight change to prompt.</p>\n        <p>Could include standard integral in Show steps (once Show steps is available)</p>\n        <p><strong>19/07/2012:</strong></p>\n        <p>Added description.</p>\n        <p>Changed Advice on the standard integral - so that it makes sense!</p>\n        <p>Added Show steps information on the standard integral.</p>\n        <p>Checked calculation.</p>\n        <p>Set new tolerance variable tol=0 for the numeric input.</p>\n        <p><strong>23/07/2012:</strong></p>\n        <p></p>\n        <p>Added tags.</p>\n        <p></p>\n        <p>Solution always requires arccosh(x) and not arcsinh(x) or arctanh(x). Is this on purpose?</p>\n        <p>&nbsp;</p>\n        <p>&nbsp;</p>\n        <p>Question appears to be working correctly.</p>\n        <p>&nbsp;</p>\n        <p><br /><br /></p>\n        <p>&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find (hyperbolic substitution):<br />$\\displaystyle \\int_{b}^{2b} \\left(\\frac{1}{\\sqrt{a^2x^2-b^2}}\\right)\\;dx$</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<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>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}