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{"name": "Integration with Partial Fractions 9", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration with Partial Fractions 9", "tags": [], "metadata": {"description": "<p>Calculating the integral of a function of the form $\\frac{mx^2+nx+k}{(x+a)(x+b)(x+c)}$ using partial fractions.</p>", "licence": "None specified"}, "statement": "<p>Calculate the integral</p>\n<p>\\[ \\simplify{int(({m}x^2+{n}x+{k})/((x+{a})(x+{b})(x+{c})),x)}. \\]</p>", "advice": "<p>In order to integrate the function \\[ \\simplify{({m}x^2+{n}x+{k})/((x+{a})(x+{b})(x+{c}))}, \\]we want to rewrite it in terms of its partial fractions.&nbsp;</p>\n<p>To do this, we want to set the function equal to the sum of 3 fractions with denominators $\\simplify{x+{a}}$, $\\simplify{x+{b}}$, and $\\simplify{x+{c}}$. Since these are all distinct linear factors, this tells us that the numerators will be constants, which we will call $A$, $B$, and $C$:</p>\n<p>\\[&nbsp;\\simplify{({m}x^2+{n}x+{k})/((x+{a})(x+{b})(x+{c}))} = \\simplify{A/(x+{a}) + B/(x+{b})+ C/(x+{c})}.\\]</p>\n<p>To find the values of $A$, $B$, and $C$, we want to first multiply this equation by the denominator of the left-hand side. This gives</p>\n<p>\\[ \\simplify{{m}x^2+{n}x+{k}=A(x+{b})(x+{c})+B(x+{a})(x+{c}) + C(x+{a})(x+{b})}.\\]</p>\n<p></p>\n<p>To find $A$, we can eliminate $B$ and $C$ by setting $x=\\var{-a}$:</p>\n<p>\\[ \\simplify{{m*a^2-n*a+k}=A{(b-a)(c-a)}} \\implies A=\\simplify[fractionNumbers]{{Asol}}.\\]</p>\n<p>Finding $B$ by setting $x=\\var{-b}$:</p>\n<p>\\[ \\simplify{{m*b^2-n*b+k}=B{(a-b)(c-b)}} \\implies B=\\simplify[fractionNumbers]{{Bsol}}.\\]</p>\n<p>Finally, setting $x=\\var{-c}$ we can find C:</p>\n<p>\\[ \\simplify{{m*c^2-n*c+k}=C{(a-c)(b-c)}} \\implies C=\\simplify[fractionNumbers]{{Csol}}.\\]</p>\n<p>Therefore,</p>\n<p>{check}</p>\n<p>Hence,</p>\n<p>\\[ \\begin{split} \\simplify{int(({m}x^2{n}x+{k})/((x+{a})(x+{b})(x+{c})),x)} &amp;\\,=&nbsp; \\simplify[all,fractionNumbers]{int({m*a^2-n*a+k}/({(b-a)*(c-a)}(x+{a}))+{m*b^2-n*b+k}/({(a-b)*(c-b)}(x+{b}))+{m*c^2-n*c+k}/({(a-c)*(b-c)}(x+{c})),x)}\\\\ \\\\ &amp;\\,=\\simplify[basic,fractionNumbers,zeroTerm,unitFactor,noLeadingMinus]{{Asol} int(1/(x+{a}),x)+{Bsol} int(1/(x+{b}),x)+{Csol} int(1/(x+{c}),x)}&nbsp; &nbsp;\\\\\\\\ &amp;\\,=\\simplify[basic,fractionNumbers,zeroTerm,unitFactor,noLeadingMinus]{{Asol} ln (abs(x+{a}))+{Bsol} ln (abs(x+{b}))+{Csol} ln (abs(x+{c})) + C}. \\end{split}\\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-6..6)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-6..6 except a)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-6..6 except [a,b])", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "if(k=1,random(-1,1)*random([1,3,4,5]),if (k=2,random(-1,1)*random([1,2,4,5]),if(k=3,random(-1,1)*random([1,2,3,5]),if(k=5,random(-1,1)*random([1,2,3,4,5,7]),random(-1,1)*random([1,2,3,4,5,7])))))", "description": "", "templateType": "anything", "can_override": false}, "Asol": {"name": "Asol", "group": "Ungrouped variables", "definition": "(m*a^2-n*a+k)/((b-a)*(c-a))", "description": "", "templateType": "anything", "can_override": false}, "Bsol": {"name": "Bsol", "group": "Ungrouped variables", "definition": "(m*b^2-n*b+k)/((a-b)*(c-b))", "description": "", "templateType": "anything", "can_override": false}, "Csol": {"name": "Csol", "group": "Ungrouped variables", "definition": "(m*c^2-n*c+k)/((a-c)*(b-c))", "description": "", "templateType": "anything", "can_override": false}, "check": {"name": "check", "group": "Ungrouped variables", "definition": "if(Asol=round(Asol) and Bsol=round(Bsol),'{sol1}','{sol2}')", "description": "", "templateType": "anything", "can_override": false}, "sol1": {"name": "sol1", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\simplify{({m}x^2+{n}x+{k})/((x+{a})(x+{b})(x+{c}))} = \\\\simplify{{Asol}/(x+{a})+{Bsol}/(x+{b})+{Csol}/(x+{c})}.\\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "sol2": {"name": "sol2", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\simplify{({n}x+{k})/((x+{a})(x+{b})(x+{c}))} = \\\\simplify[all,fractionNumbers]{{m*a^2-n*a+k}/({(a-b)(a-c)}(x+{a}))+{m*b^2-n*b+k}/({(b-a)(b-c)}(x+{b}))+{m*c^2-n*c+k}/({(c-a)(c-b)}(x+{c}))} .\\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random([1,2,3,5,7])", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", 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