Integrate the following function $f(x)$.
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Input the constant of integration as $C$.
2/08/2012:
\n\t\tAdded tags.
\n\t\tAdded description.
\n\t\tChecked calculation. OK.
\n\t\tAdded decimal point to forbidden strings along with message to user re input of numbers.
\n\t\tMessage about Show steps included. Also another message about including the constant of integration.
\n\t\tChanged checking range from 0 to 1 to 1 to 2.
\n\t\tImproved display.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\int ax ^ m+ bx^{c/n}\\;dx$.
"}, "ungrouped_variables": ["a", "c", "b", "d", "s1", "m", "n", "r"], "parts": [{"prompt": "\n\t\t\t$\\simplify[std]{f(x) = {c}x ^ {m} + {d}*x^({b}/{n})}$
\n\t\t\t$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]
\n\t\t\tInput all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.
\n\t\t\tClick on Show steps to get more information. You will not lose any marks by doing so.
\n\t\t\t", "type": "gapfill", "marks": 0, "showCorrectAnswer": true, "gaps": [{"checkvariablenames": false, "showCorrectAnswer": true, "answer": "({c}/{m+1})x ^ {m+1} + ({d*n}/{b+n})*x^({n+b}/{n})+C", "vsetrange": [1, 2], "answersimplification": "std", "expectedvariablenames": [], "vsetrangepoints": 5, "type": "jme", "scripts": {}, "checkingtype": "absdiff", "notallowed": {"partialCredit": 0, "message": "Input all numbers as integers or fractions and not decimals.
", "showStrings": false, "strings": ["."]}, "marks": 3, "checkingaccuracy": 0.001, "showpreview": true}], "stepsPenalty": 0, "scripts": {}, "steps": [{"prompt": "The indefinite integral of a power $x^n$ where $n\\neq -1$ is \\[\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C\\]
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\\[\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C\\] for any number $n \\neq -1$ we have
\\[\\begin{eqnarray*}\n\t \n\t \\simplify[std]{Int({c}*x^{m}+{d}*x ^ ({b} / {n}),x)} &=&\\simplify[std]{ ({c} / {m + 1}) * x ^ {m + 1} +{d}* x ^ ({b} / {n} + 1) / ({b} / {n} + 1) + C }\\\\\n\t \n\t &=&\\simplify[std]{ ({c} / {m + 1}) * x ^ {m + 1} + ({d*n} / {b + n}) * x ^ ({b + n} / {n}) + C}\n\t \n\t \\end{eqnarray*}\\]