2005 TYS

The indefinite integral $\int{\frac{\mathrm{P}(x)}{{{x}^{3}}+1}\,\mathrm{d}x}$, where $\mathrm{P}\left( x \right)$ is a polynomial in $x$, is denoted by $\mathrm{I}$.

1. Find $\mathrm{I}$ when $\mathrm{P}(x)={{x}^{2}}$.

   [2]

2. By writing ${{x}^{3}}+1=\left( x+1 \right)\left( {{x}^{2}}+Ax+B \right)$, where $A$ and $B$ are constants, find $\mathrm{I}$ when
   1. $\mathrm{P}\left( x \right)={{x}^{2}}-x+1$,

      [3]

   2. $\mathrm{P}\left( x \right)=x+1$.

      [3]

3. Using the results of parts (i) and (ii), or otherwise, find $\mathrm{I}$ when $\mathrm{P}\left( x \right)=1$.

   [4]