<ol class="abc-list"><li><div class="list-item-paragraph">Find the following integrals:</div><ol class="i-list"><li><div class="list-item-paragraph">$\displaystyle\int{\left( {{\cos }^{4}}x-{{\sin }^{4}}x \right)}\mathrm{ d}x$;</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">$\displaystyle\int{{{\log }_{3}}}\left( 3x-1 \right)\mathrm{ d}x$.</div><p style="text-align: right;">[4]</p></li></ol></li><li><div class="list-item-paragraph">By using the substitution $x=4\sin \theta $, show that</div><div class="list-item-paragraph" style="text-align: center">$\displaystyle\int{\frac{\sqrt{16-{{x}^{2}}}}{{{x}^{2}}}}\mathrm{ d}x=-\frac{\sqrt{16-{{x}^{2}}}}{x}-{{\sin }^{-1}}\left( \frac{x}{4} \right)+C$.</div><div class="list-item-paragraph">where $C$ is an arbitrary constant.</div><p style="text-align: right;">[5]</p></li></ol>

By using the substitution $x=4\sin \theta $, show that

$\displaystyle\displaystyle \displaystyle\int{\frac{\sqrt{16-{{x}^{2}}}}{{{x}^{2}}}}\mathrm{ d}x=-\frac{\sqrt{16-{{x}^{2}}}}{x}-{{\sin }^{-1}}\left( \frac{x}{4} \right)+C$.

2012 HCI P1 Q10

2012 HCI P1 Q10

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