Question

Thanks for helping with calculus

Answer 1

Integrating $f(x)$ gives

$g(x)=\displaystyle\int f(x)\,\mathrm dx=C+\sum_{n=0}^\infty\frac{a_n}{n+1}x^{n+1}$

Notice that $x=0$ forces all terms in the sum to be 0, so that

$g(0)=C=5$

and hence

$g(x)=\displaystyle5+\sum_{n=0}^\infty\frac{a_n}{n+1}x^{n+1}$

making the first option correct.