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{"name": "Delvis integrasjon", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Delvis integrasjon", "tags": [], "metadata": {"description": "", "licence": "All rights reserved"}, "statement": "<p>I denne oppgaven regner vi ut det ubestemte integralet&nbsp;</p>\n<p>\\[ \\int (\\simplify{{b}x+{c}})\\cdot e^{x}dx\\]</p>\n<p>steg for steg ved bruk av delvis integrasjon.&nbsp;</p>\n<p>Setningen som vi skal bruke sier at for funksjonene $u(x)$ og $v(x)$ som har kontinuerlige deriverte gjelder formelen</p>\n<p>\\[\\int u(x)\\cdot v'(x)dx=u(x)\\cdot v(x) -\\int u'(x)\\cdot v(x) dx&nbsp; \\]</p>", "advice": "<p>F&oslash;rst, er funksjonene $u(x)=\\simplify{{b}x+{c}}$ og $v'(x)=e^x$</p>\n<p>Videre regner ut $u'(x)=\\var{b}$ og $v(x)=e^x$</p>\n<p>N&aring; bruker setningen og skriver det opprinnelige integralet som differansen og regner ut</p>\n<p>\\[\\int (\\simplify{{b}x+{c}})\\cdot e^{x}dx=(\\simplify{{b}x+{c}})\\cdot e^{x}-\\int \\var{b} e^x dx=(\\simplify{{b}x+{c-b}})e^x+C\\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except 1)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except -b)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["b", "c"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, 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