Integrating [tex]f(x)[/tex] gives
[tex]g(x)=\displaystyle\int f(x)\,\mathrm dx=C+\sum_{n=0}^\infty\frac{a_n}{n+1}x^{n+1}[/tex]
Notice that [tex]x=0[/tex] forces all terms in the sum to be 0, so that
[tex]g(0)=C=5[/tex]
and hence
[tex]g(x)=\displaystyle5+\sum_{n=0}^\infty\frac{a_n}{n+1}x^{n+1}[/tex]
making the first option correct.
