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{"name": "Vinitha's copy of Integration by 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", "c", "b", "s1"], "name": "Vinitha's copy of Integration by substitution", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "\n  <p>This exercise is best solved by using substitution.</p>\n  <p>Note that the numerator $\\simplify[std]{{2 * a} * x + {b}}$ of \\[\\simplify[std]{({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c})}\\] is the derivative of the denominator $\\simplify[std]{{a} * x ^ 2 + {b} * x + {c}}$</p>\n  <p>So if you use as your substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$ you then have $\\simplify[std]{ du = ({2 * a} * x + {b}) * dx}$</p>\n  <p>Hence we can replace $\\simplify[std]{ ({2 * a} * x + {b}) * dx}$ by $du$</p>\n  <p>Hence the integral becomes:</p>\n  <p>\\[\\begin{eqnarray*} I&amp;=&amp;\\int\\;\\frac{du}{u}\\\\ &amp;=&amp;\\ln(|u|)+C\\\\ &amp;=&amp; \\simplify[std]{ln(abs({a} * (x ^ 2) + ({b} * x) + {c}))+C} \\end{eqnarray*}\\]<br/><br/></p>\n  <p><strong>A Useful Result</strong><br/>This example can be generalised.</p>\n  <p>Suppose \\[I = \\int\\; \\frac{f'(x)}{f(x)}\\;dx\\]</p>\n  <p>The using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:<br/>\\[I = \\int \\frac{du}{u} = \\ln(|u|)+ C = \\ln(|f(x)|)+C\\]</p>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "parts": [], "extensions": [], "statement": "\n  <p>Find the following integral.</p>\n  <p>You must input the constant of integration as $C$.</p>\n  <p>Input all numbers as integers or fractions.</p>\n  <p>You can click on Show steps to get a hint. You will lose 1 mark if you do so.</p>\n  <p>Note that $\\displaystyle \\int \\frac{1}{x}\\;dx=\\ln(|x|)+C$ and you must include the absolute value in the argument of $\\ln$. You input $|x|$ as abs(x).</p>\n  ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "ceil(b^2/4a+random(1..5))", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "s1*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}}, "metadata": {"description": "<p>Find $\\displaystyle I=\\int \\frac{2 a x + b} {a x ^ 2 + b x + c}\\;dx$ by substitution or otherwise.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Vinitha Sebastine", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/814/"}]}]}], "contributors": [{"name": "Vinitha Sebastine", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/814/"}]}