Question 819248
The phrase "most general" must mean that you need to add a constant.
 
a.) The derivative of {{{5x^4+4x^3+4x}}} is {{{f(x)=20x^3+12x^2+4}}} ,
but the derivative of {{{highlight(5x^4+4x^3+4x+K)}}} where the constant {{{K}}} represents any number is {{{f(x)}}} too.
 
b.) You wrote g(x)=3x^4-x^3+6x^2/x^4={{{3x^4-x^3+6x^2/x^4=3x^4-x^3+6x^(-2)}}}
The antiderivative of that would be
{{{3x^5/5-x^4/4+6(x^(-1)/(-1))+K=3x^5/5-x^4/4+K-6/x}}}
However, if you meant g'(x)=(3x^4-x^3+6x^2)/x^4={{{(3x^4-x^3+6x^2)/x^4=3-x^(-1)+6x^(-2)=3x-1/x+6/x^2}}} ,
the antiderivative is
{{{3x-ln(x)+6(x^(-1)/(-1))+K=3x-ln(x)-6/x+K}}}
 
c.) If you meant h(x)={{{e^x+sin(x)+14root(5,x^2)=e^x+sin(x)+14x^"2/5"}}},
The antiderivative of that function can be calculated as
{{{e^x+sin(x)+14(x^("2/5"+1)/("2/5"+1))=e^x+sin(x)+14x^"7/5"/(7/5)=e^x+sin(x)+14(5/7)x^"7/5"=e^x+sin(x)+10root(5,x^7)}}}