// Numbas version: finer_feedback_settings
{"name": "Integration with Partial Fractions 10", "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 10", "tags": [], "metadata": {"description": "<p>Calculating the integral of a function of the form $\\frac{mx^2+nx+k}{(x+a)(x+b)^2}$ using partial fractions.</p>", "licence": "None specified"}, "statement": "<p>Calculate the following integral:</p>\n<p>\\[ \\simplify{int(({m}x^2+{n}x+{k})/((x+{a})(x+{b})^2),x)}. \\]</p>", "advice": "<p>In order to integrate the function \\[ \\simplify{({m}x^2+{n}x+{k})/((x+{a})(x+{b})^2)}, \\] we want to rewrite it in terms of its partial fractions.</p>\n<p>Since we have a distinct linear factor and a repeated linear factor, we want to set the function equal to the sum of 3 fractions with denominators $\\simplify{x+{a}}$, $\\simplify{x+{b}}$, and $\\simplify{(x+{b})^2}$. 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})^2)} = \\simplify{A/(x+{a}) + B/(x+{b})+ C/(x+{b})^2}.\\]</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})^2+B(x+{a})(x+{b}) + C(x+{a})}.\\]</p>\n<p><strong>(Note: To find $A$, $B$, and $C$, we will use a combination of choosing suitable values of $x$ to eliminate terms, and equating coefficients. It can be solved by <em>only</em> equating coefficients, but this is a more efficient process.)</strong></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)^2}} \\implies A=\\simplify[fractionNumbers]{{Asol}}.\\]</p>\n<p>To find $C$, we can eliminate $A$ and $B$ by setting $x=\\var{-b}$:</p>\n<p>\\[ \\simplify{{m*b^2-n*b+k}=C{(a-b)}} \\implies C=\\simplify[fractionNumbers]{{Csol}}.\\]</p>\n<p>Finally, by equating coefficients of the $x^2$-terms we can find $B$:</p>\n<p>\\[ (x^2): \\quad \\var{m} = \\simplify{A+B} \\implies B=\\var{m}-A. \\]</p>\n<p>Therefore, \\[ B=\\simplify[fractionNumbers]{{Bsol}}, \\]</p>\n<p>and</p>\n<p>{check}</p>\n<p>Hence,</p>\n<p>{check2}</p>\n<p></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}, "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": "rational(((m*a^2-n*a+k)/(b-a))/(b-a))", "description": "", "templateType": "anything", "can_override": false}, "Bsol": {"name": "Bsol", "group": "Ungrouped variables", "definition": "rational((m*b^2-2*m*a*b+n*a-k)/(b-a)^2)", "description": "", "templateType": "anything", "can_override": false}, "Csol": {"name": "Csol", "group": "Ungrouped variables", "definition": "rational((m*b^2-n*b+k)/(a-b))", "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})^2)} = \\\\simplify{{Asol}/(x+{a})+{Bsol}/(x+{b})+{Csol}/(x+{b})^2}.\\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "sol2": {"name": "sol2", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\simplify{({m}x^2+{n}x+{k})/((x+{a})(x+{b})^2)} = \\\\simplify[all,fractionNumbers]{{m*a^2-n*a+k}/({(b-a)^2}(x+{a}))+{m*b^2-2*m*a*b+n*a-k}/({(b-a)^2}(x+{b}))+{m*b^2-n*b+k}/({(a-b)}(x+{b})^2)}.\\\\]</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": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}, "check2": {"name": "check2", "group": "Ungrouped variables", "definition": "if(Asol=round(Asol),'{int1}','{int2}')", "description": "", "templateType": "anything", "can_override": false}, "int1": {"name": "int1", "group": "Ungrouped variables", "definition": "\"<p>\\\\[\\\\begin{split} \\\\simplify{int(({m}x^2+{n}x+{k})/((x+{a})(x+{b})^2),x)} &amp;\\\\,= \\\\simplify[all,fractionNumbers]{int({Asol}/(x+{a})+{Bsol}/(x+{b})+{Csol}/(x+{b})^2,x)} \\\\\\\\\\\\\\\\ &amp;\\\\,=\\\\simplify[all,!collectLikeFractions,fractionNumbers,zeroTerm,unitFactor,noLeadingMinus]{{Asol} int(1/(x+{a}),x)+{Bsol} int(1/(x+{b}),x)+{Csol} int((x+{b})^-2,x)} \\\\\\\\\\\\\\\\ &amp;\\\\,=\\\\simplify[basic,fractionNumbers,zeroTerm,unitFactor,noLeadingMinus]{{Asol} ln(abs(x+{a}))+{Bsol} ln(abs(x+{b}))-{Csol}/(x+{b})+C}.\\\\end{split}\\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "int2": {"name": "int2", "group": "Ungrouped variables", "definition": "\"<p>\\\\[\\\\begin{split} \\\\simplify{int(({m}x^2+{n}x+{k})/((x+{a})(x+{b})^2),x)} &amp;\\\\,= \\\\simplify[all,fractionNumbers]{int({m*a^2-n*a+k}/({(b-a)^2}(x+{a}))+{m*b^2-2*m*a*b+n*a-k}/({(b-a)^2}(x+{b}))+{m*b^2-n*b+k}/({a-b}(x+{b})^2),x)} \\\\\\\\\\\\\\\\ &amp;\\\\,=\\\\simplify[all,!collectLikeFractions,fractionNumbers,zeroTerm,noLeadingMinus]{{Asol} int(1/(x+{a}),x)+{Bsol} int(1/(x+{b}),x)+{Csol} int((x+{b})^-2,x)} \\\\\\\\\\\\\\\\ &amp;\\\\,=\\\\simplify[basic,fractionNumbers,zeroTerm,noLeadingMinus]{{Asol} ln(abs(x+{a}))+{Bsol} ln(abs(x+{b}))-{Csol}/(x+{b})}+C.\\\\end{split}\\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}}, 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