<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $\int{\frac{6\tan 3x}{1+\cos 6x}\mathrm{d}x}$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">It is given that ${{I}_{n}}=\int_{1}^{\mathrm{e}}{{{\left( \ln x \right)}^{n}}\mathrm{d}x}$, for $n\in {{\mathbb{Z}}^{+}}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Use integration by parts to show that</div><div class="list-item-paragraph" style="text-align: center">${{I}_{n}}=\mathrm{e}-n{{I}_{n-1}}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">The region bounded by the curve $y={{\left( \ln x \right)}^{2}}$, the $x$-axis and the line $x=\mathrm{e}$ is rotated through $2\pi $ radians about the $x$-axis. Using the result in <strong>(b)(i)</strong>, find the exact volume of the solid formed.</div><p style="text-align: right;">[3]</p></li></ol></li></ol>

Find $\displaystyle\displaystyle \int{\frac{6\tan 3x}{1+\cos 6x}\mathrm{d}x}$.

It is given that ${{I}_{n}}=\displaystyle\displaystyle \int_{1}^{\mathrm{e}}{{{\left( \ln x \right)}^{n}}\mathrm{d}x}$, for $n\in {{\mathbb{Z}}^{+}}$.

The region bounded by the curve $y={{\left( \ln x \right)}^{2}}$, the $x$-axis and the line $x=\mathrm{e}$ is rotated through $2\pi $ radians about the $x$-axis. Using the result in (b)(i), find the exact volume of the solid formed.

2021 HCI P1 Q4

2021 HCI P1 Q4

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