Question 1207322
<pre>
{{{"f(x)"=a[0]x^n+a[1]x^(n-1)+""*""*""*""+a[n-1]x+a[n]}}}

{{{"f'(x)"=n*a[0]x^(n-1)+(n-1)*a[1]x^(n-2)+""*""*""*""+a[n-1]}}}

{{{int("f(x)",dx)}}}{{{""=""}}}{{{expr(1/(n+1))a[0]x^(n+1)+expr(1/n)a[1]x^n+""*""*""*""+expr(1/2)a[n-1]x^2+a[n]x+C}}}

Since the sum of those must be identically equal to {{{x^4+13x^2+2}}},

then the largest power of x that occurs in the sum, which is n+1, must be
the largest power that occurs in {{{x^4+13x^2+2}}}, which is 4. Therefore,
n+1 = 4 and n=3.  Substituting (and eliminating the 3 dots since they
are no longer needed):

{{{"f(x)"=a[0]x^3+a[1]x^(3-1)+a[3-1]x+a[3]}}}
{{{"f'(x)"=3*a[0]x^(3-1)+(3-1)*a[1]x^(3-2)+a[3-1]}}}
{{{int("f(x)",dx)}}}{{{""=""}}}{{{expr(1/(3+1))a[0]x^(3+1)+expr(1/3)a[1]x^3+expr(1/2)a[3-1]x^2+a[3]x+C}}}

Simplifying,

{{{"f(x)"=a[0]x^3+a[1]x^2+a[2]x+a[3]}}}
{{{"f'(x)"=3*a[0]x^2+2*a[1]x+a[2]}}}
{{{int("f(x)",dx)}}}{{{""=""}}}{{{expr(1/4)a[0]x^4+expr(1/3)a[1]x^3+expr(1/2)a[2]x^2+a[3]x+C}}}

Since the coefficients of x<sup>3</sup> and x are 0 in {{{x^4+13x^2+2}}},

{{{a[0]x^3+expr(1/3)*a[1]x^3=0}}} and {{{a[2]x+2a[1]x+a[3]=0}}}.

Since the coefficient of x<sup>2</sup> is 13, {{{a[1]x^2+3a[0]x^2+expr(1/2)a[2]x^2=13}}} 

and since its constant term is 2, {{{a[3]+a[2]+C=2}}}

So we have the system of equations

{{{system(
a[0]+expr(1/3)*a[1]=0,
a[1]+3*a[0]+expr(1/2)*a[2]=13,
a[2]+2*a[1]+a[3]=0,
a[3]+a[2]+C=2)}}}

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: 

{{{system(a(0)=-1/3,a(1)=1,a(2)=26,a(3)=-26)}}}

So that makes

{{{"f(x)"=expr(-1/3)x^3+x^2+26x-26}}}

Edwin</pre>