SOLUTION: f (x) is polynomial function, f '(x) + int f (x) dx = x ^4 + 13 x ^2 + 2. Find f (x)

Since the sum of those must be identically equal to ,

then the largest power of x that occurs in the sum, which is n+1, must be
the largest power that occurs in , which is 4. Therefore,
n+1 = 4 and n=3.  Substituting (and eliminating the 3 dots since they
are no longer needed):





Simplifying,





Since the coefficients of x3 and x are 0 in ,

 and .

Since the coefficient of x2 is 13,  

and since its constant term is 2, 

So we have the system of equations



Your teacher must have meant for you to take the constant 
of integration to be 0.  Otherwise you have 5 unknowns but 
only 4 equations.  I'll use my system solver to solve that.  
I get: 



So that makes



Edwin

We want to find f(x) as a polynomial  f(x) =  +  + . . . + .

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)     =                         + 2bx + c,

    int f(x) dx =   +  +  + dx  + E.



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

    (a)  coefficient at    is    It gives an equation

              = 1;   hence  a = 4.



    (b)  coefficient at    is  0.  It gives an equation

              = 0;   hence  b = 0.



    (c)  coefficient at    is  13.  It gives an equation

              = 13,  or   = 13  --->  = 13 - 12 = 1  --->  c = 2.



    (d)  coefficient at    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.     ANSWER



CHECK.  The derivative is f ' (x) = .

        The anti-derivative is  F(x) =   = .

        The sum f ' (x) + F(x) =  +  = .   ! correct !