// Numbas version: finer_feedback_settings {"name": "Musa's copy of 3 Integration: Indefinite integral", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Musa's copy of 3 Integration: Indefinite integral", "tags": [], "metadata": {"description": "

Find $\\displaystyle \\int ae ^ {bx}+ c\\sin(dx) + px ^ {q} + k/x \\;dx$.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\n\t

Integrate the following function $f(x)$.

\n\t

Input the constant of integration as $C$.

\n\t", "advice": "\n\t

Note that \\[\\begin{eqnarray*} &\\int& \\;x^n\\;dx&=&\\frac{x^{n+1}}{n+1}+C,\\;\\;n \\neq -1\\\\ &\\int& \\;\\sin(ax)\\;dx &=& -\\frac{1}{a}\\cos(ax)+C\\\\ &\\int& \\;e^{ax}\\;dx &=& \\frac{1}{a}e^{ax}+C\\\\ \\end{eqnarray*}\\]

\n\t

Splitting the integral into three parts and using the above information we have:
\\[\\begin{eqnarray*}\\simplify[std]{Int({b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3},x)}&=&\\simplify[std]{Int({b} * e ^ ({a}*x),x)+Int({b1} * Sin({a1}*x),x)+Int({a2} * x ^ {c3},x) }\\\\ &=&\\simplify[std]{({b}/{a}) * (e ^({a}*x)) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+C} \\end{eqnarray*}\\]

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$\\simplify[std]{f(x) = {a5}* x^{an5}+ {b5}*x^{bn5}+ {c5}}$

$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]

Input all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.

Click on Show steps to get more information. You will not lose any marks by doing so.

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Note that \\[\\begin{eqnarray*} &\\int& \\;x^n\\;dx&=&\\frac{x^{n+1}}{n+1}+C\\\\ \\end{eqnarray*}\\]

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Input all numbers as integers or fractions and not decimals.

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$\\simplify[std]{f(x) = {b6} * e ^ ({a6}*x) + {c6}}$

$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]

Input all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.

Click on Show steps to get more information. You will not lose any marks by doing so.

Note that \\[\\begin{eqnarray*} &\\int& \\;e^{ax}\\;dx &=& \\frac{1}{a}e^{ax}+C \\end{eqnarray*}\\]

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Input all numbers as integers or fractions and not decimals.

$\\simplify[std]{f(x) = {b7} * Sin({a7}*x) + {b8} * Cos({a8}*x) }$

$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]

Input all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.

Click on Show steps to get more information. You will not lose any marks by doing so.

Note that \\[\\begin{eqnarray*} &\\int& \\;\\sin(ax)\\;dx &=& -\\frac{1}{a}\\cos(ax)+C\\\\ &\\int& \\;\\cos(ax)\\;dx &=& \\frac{1}{a}\\sin(ax)+C\\\\ \\end{eqnarray*}\\]

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Input all numbers as integers or fractions and not decimals.

$\\simplify[std]{f(x) = ({a9} /x) + ({b9} /(x^2)) }$ (use abs(x) for |x|)

$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]

Input all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.

Click on Show steps to get more information. You will not lose any marks by doing so.

Note that \\[\\begin{eqnarray*} &\\int& \\;\\frac{1}{x}\\;dx&=&\\ln(|x|)+C, \\,\\, {\\rm and} \\,\\, \\frac{1}{x^2}&=x^{-2}& \\end{eqnarray*}\\]

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Input all numbers as integers or fractions and not decimals.

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