Integrate the given series expansion of $\displaystyle {f}$  term-by-term from zero to $\displaystyle {x}$  to obtain the corresponding series expansion for the indefinite integral of $\displaystyle {f}$.

$\displaystyle {f{{\left({x}\right)}}}=\frac{{-{2}{x}}}{{\left({1}+{x}^{{2}}\right)}^{{2}}}={\sum_{{{n}={0}}}^{{\infty}}}{\left(-{1}\right)}^{{n}}{2}{n}{x}^{{{2}{n}-{1}}}$

$\displaystyle {\int_{{0}}^{{x}}}{f{{\left({t}\right)}}}{\left.{d}{t}\right.}={\sum_{{{n}={1}}}^{{\infty}}}$ Preview Question 1