1. $\\displaystyle \\int_0^\\infty\\;e^{-\\var{a}x}\\,dx=\\; $[[0]]
Input your answer as a fraction.
2. $\\displaystyle \\int_1^2\\;\\frac{1}{x^\\var{a1}}\\,dx=\\;$[[1]]
Input your answer as a fraction.
3. $\\displaystyle \\int_0^{\\pi}\\;\\cos\\left(\\frac{x}{\\var{2*n}}\\right)\\,dx=\\;$[[2]]
Input your answer to 3 decimal places.
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", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "{p}/{q}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "answersimplification": "std", "expectedvariablenames": [], "notallowed": {"message": "Input your answer as a fraction
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "reldiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "val+tolerance", "minValue": "val-tolerance", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "Evaluate the following definite integrals.
", "tags": ["calculus", "Calculus", "checked2015", "definite integral", "definite integration", "exponential function", "integration", "integration of a negative power", "integration of a negative power ", "integration of an exponential", "integration of trigonometric functions", "mas1601", "MAS1601", "trigonometric functions"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t20/06/2012:
\n \t\tAdded tags.
\n \t\tImproved display of prompts.
\n \t\tIncomplete loading of question into editor noted. OK if refreshed.
\n \t\tChanged accuracy for second question to relative difference of 0.0001 to ensure it marked correctly for extreme values.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Calculate definite integrals: $\\int_0^\\infty\\;e^{-ax}\\,dx$, $\\int_1^2\\;\\frac{1}{x^{b}}\\,dx$, $\\; \\int_0^{\\pi}\\;\\cos\\left(\\frac{x}{2n}\\right)\\,dx$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n1. We have
\\[\\int\\;e^{-\\var{a}x}\\,dx=-\\frac{1}{\\var{a}}e^{-\\var{a}x} +C\\]
Also we know that $\\lim_{x \\to \\infty}\\;e^{-\\var{a}x}=0$.
Hence:
\\[\\begin{eqnarray*} \\int_0^\\infty\\;e^{-\\var{a}x}\\,dx&=&-\\frac{1}{\\var{a}}\\left[e^{-\\var{a}x}\\right]_0^\\infty\\\\ &=&-\\frac{1}{\\var{a}}\\left(\\left(\\lim_{x \\to \\infty}e^{-\\var{a}x}\\right)-1\\right)\\\\ &=&-\\frac{1}{\\var{a}}(0-1)\\\\ &=&\\frac{1}{\\var{a}} \\end{eqnarray*} \\]
2. \\[\\int\\;\\frac{1}{x^\\var{a1}}\\,dx= \\simplify[std]{x^{-a1+1}/{-a1+1}}+C\\]
Hence:
\\[ \\begin{eqnarray*}\\int_1^2\\;\\frac{1}{x^\\var{a1}}\\,dx&=&\\left[\\simplify[std]{x^{-a1+1}/{-a1+1}}\\right]_1^2\\\\&=& -\\simplify[std]{{1}/{a1-1}}\\left(2^{\\var{-a1+1}}-1\\right)\\\\&=&\\simplify[std]{{p}/{q}} \\end{eqnarray*} \\]
3. \\[ \\begin{eqnarray*} \\int_0^{\\pi}\\;\\cos\\left(\\frac{x}{\\var{2*n}}\\right)\\,dx&=&\\var{2*n}\\left[\\sin\\left(\\frac{x}{\\var{2*n}}\\right)\\right]_0^{\\pi}\\\\ &=&\\var{2*n}\\left(\\sin\\left(\\frac{\\pi}{\\var{2*n}}\\right)-0\\right)\\\\ &=&\\var{val} \\end{eqnarray*} \\]
to 3 decimal places.