Question 1207322
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f (x) is  {{{highlight(highlight(a))}}}  polynomial function, f '(x) + int f (x) dx = x ^4 + 13 x ^2 + 2, 
Find  f(x)
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<pre>
We want to find f(x) as a polynomial  f(x) = {{{a[0]*x^n}}} + {{{a[1]*x^(n-1)}}} + . . . + {{{a[n]}}}.

Taking derivative decreases the degree of a polynomial by one unit.
Taking antiderivative increases the degree of a polynomial by one unit.


Since the sum   f ' (x) + int f (x) dx   is a polynomial of degree 4,
                ----------------------

it means that the sough polynomial f(x) is of degree 3:

    f(x) = ax^3 + bx^2 + cx + d.


Then 

    f ' (x)     =                        {{{3a*x^2}}} + 2bx + c,

    int f(x) dx =  {{{(a/4)x^4}}} + {{{(b/3)x^3}}} + {{{(c/2)x^2}}} + dx  + E.



So, in the sum   f ' (x) + int f(x) dx
                 ----------------------

    (a)  coefficient at  {{{x^4}}}  is  {{{a/4}}}  It gives an equation

             {{{a/4}}} = 1;   hence  a = 4.



    (b)  coefficient at  {{{x^3}}}  is  0.  It gives an equation

             {{{b/3}}} = 0;   hence  b = 0.



    (c)  coefficient at  {{{x^2}}}  is  13.  It gives an equation

             {{{3a+ c/2}}} = 13,  or  {{{3*4 + c/2}}} = 13  ---> {{{c/2}}} = 13 - 12 = 1  --->  c = 2.



    (d)  coefficient at  {{{x}}}  is  0.  It gives an equation

             2b + d = 0,  which implies  2*0 + d = 0;  hence,  d = 0.



    +------------------------------------------------------------+
    |    At this point, the problem is just solved to the end.   |
    |            a = 4;  b = 0;  c = 2;  d = 0.                  |
    +------------------------------------------------------------+



The sough polynomial is  f(x) = 4x^3 + 2x.     <U>ANSWER</U>



<U>CHECK</U>.  The derivative is f ' (x) = {{{12x^2 + 2}}}.

        The anti-derivative is  F(x) =  {{{(4/4)x^4 + (2/2)x^2}}} = {{{x^4 + x^2}}}.

        The sum f ' (x) + F(x) = {{{(12x^2+2)}}} + {{{(x^4 + x^2)}}} = {{{x^4 + 13x^2 + 2}}}.   ! correct !
</pre>

Solved.


Do not accept any other &nbsp;{{{highlight(different)}}} &nbsp;answer.



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The solution by &nbsp;Edwin is &nbsp;INCORRECT.


To make sure that it is incorrect, &nbsp;simply take the antiderivative of his leading term  &nbsp;{{{(-1/3)x^3}}}.


This antiderivative is  &nbsp;{{{(-1/(3*4))*x^4}}} = {{{(-1/12)x^4}}}, &nbsp;and no other arguments are needed anymore.