Recovering\noriginal function given some information such as derivative and value at some point.
Find the orginal function $f(x)$ given $f^\\prime (x)$ and value $f(x_0) = C_0;$ that is solve for constant for $\\int f^\\prime (x) \\,dx.$
", "advice": "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\nFirst integrate:
\n(*) $\\int f^\\prime (x)\\,dx$ = $\\int\\;(a x^n+ c)\\,dx=\\; \\frac{a}{n+1}x^{n+1} + cx +C$
\nthen calculate the value of $C$ from
\n$\\frac{a}{n+1}x_0^{n+1} + cx_0 +C = C_0$
\nand put in back to (*).
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\n$\\displaystyle \\int\\;(\\simplify[std]{ {a5}* x^{an5}+ {c5}})\\,dx=\\;$[[0]]
\n$f (x)=\\;$[[1]]
\nInput all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.
\nClick on Show steps to get more information. You will not lose any marks by doing so.
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\n$\\displaystyle \\int\\;(\\simplify[std]{ {a} sin({an}x)+ {c}})\\,dx=\\;$[[0]]
\n$f(x)=\\;$[[1]]
\nInput all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.
\nPut 'pi' for $\\pi$ and do not use decimals.
\nClick on Show steps to get more information. You will not lose any marks by doing so.
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