<p>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 $\mathrm{I}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find $\mathrm{I}$ when $\mathrm{P}(x)={{x}^{2}}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">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</div><ol class="abc-list"><li><div class="list-item-paragraph">$\mathrm{P}\left( x \right)={{x}^{2}}-x+1$,</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">$\mathrm{P}\left( x \right)=x+1$.</div><p style="text-align: right;">[3]</p></li></ol></li><li><div class="list-item-paragraph">Using the results of parts <strong>(i)</strong> and <strong>(ii)</strong>, or otherwise, find $\mathrm{I}$ when $\mathrm{P}\left( x \right)=1$.</div><p style="text-align: right;">[4]</p></li></ol>

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

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

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

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

N2005 P1 Q14 [Either]

N2005 P1 Q14 [Either]

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