// Numbas version: exam_results_page_options {"name": "Henrik Skov's copy of Calculate definite integral with a hyperbolic substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "tol", "val", "valacc"], "name": "Henrik Skov's copy of Calculate definite integral with a hyperbolic substitution", "tags": ["arccosh", "calculus", "Calculus", "checked2015", "cr1", "definite integration", "hyperbolic functions", "integration", "integration by substitution", "inverse hyperbolic functions", "MAS1601", "standard integrals", "Steps", "steps", "substitution", "tested1"], "advice": "

\\[I=\\int_{\\var{b}}^{\\var{2*b}} \\left(\\frac{1}{\\sqrt{\\simplify[std]{{a^2}x^2-{b^2}}}}\\right)\\;dx\\]

To solve this, we will employ the standard integral:\\[\\int \\frac{1}{\\sqrt{x^2-1}}\\;dx=\\simplify{arccosh(x)+C}\\]

We 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:

\\[\\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*} \\]

where 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 $v$):

\\[\\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.

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First, specify the answer to $\\displaystyle{\\int \\frac{1}{\\sqrt{x^2-1}}dx}$:

", "matrix": [0, 1, 0], "shuffleChoices": false, "scripts": {}, "choices": ["$\\textrm{arcsinh}(x)+C$", "$\\textrm{arccosh}(x)+C$", "$\\textrm{arctanh}(x)+C$"], "marks": 0, "displayType": "radiogroup", "maxMarks": 1, "distractors": ["", "", ""], "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "minMarks": 0}, {"prompt": "

Use your answer from above to evaluate the required integral.

$I=\\;\\;$[[0]]

Input your answer to $2$ decimal places.

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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\\]

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "random(2..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "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": ""}, "val": {"definition": "precround(valacc,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "val", "description": ""}, "tol": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}}, "metadata": {"notes": "

30/06/2012:

Added, edited tags

Slight change to prompt.

Could include standard integral in Show steps (once Show steps is available)

19/07/2012:

Added description.

Changed Advice on the standard integral - so that it makes sense!

Added Show steps information on the standard integral.

Checked calculation.

Set new tolerance variable tol=0 for the numeric input.

23/07/2012:

Added tags.

Solution always requires arccosh(x) and not arcsinh(x) or arctanh(x). Is this on purpose?

Question appears to be working correctly.

22/12/2012:(WHF)

Checked calculation, OK. Added tested1 tag.

Checked rounding, OK. Added cr1 tag.

", "description": "

Find (hyperbolic substitution):
$\\displaystyle \\int_{b}^{2b} \\left(\\frac{1}{\\sqrt{a^2x^2-b^2}}\\right)\\;dx$

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