<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Without using a graphing calculator, show that</span></div><div class="list-item-paragraph" style="text-align: center">$\int_{0}^{1}{\frac{{{x}^{3}}}{1+{{x}^{2}}}}\mathrm{ d}x=k\left( 1-\ln 2 \right)$,</div><div class="list-item-paragraph">where $k$ is a real number to be determined.</div><p style="text-align: right;">[3]</p><div class="list-item-paragraph">Hence find $\int_{0}^{1}{{{x}^{2}}{{\tan }^{-1}}x}\mathrm{ d}x$, leaving your answer in exact form.</div><p style="text-align: right;">[3]</p></li></ol>

Without using a graphing calculator, show that

$\displaystyle\displaystyle \int_{0}^{1}{\frac{{{x}^{3}}}{1+{{x}^{2}}}}\mathrm{ d}x=k\left( 1-\ln 2 \right)$,

where $k$ is a real number to be determined.

Hence find $\displaystyle\displaystyle \int_{0}^{1}{{{x}^{2}}{{\tan }^{-1}}x}\mathrm{ d}x$, leaving your answer in exact form.

2018 TJC Promo Q5

2018 TJC Promo Q5

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