Find $\displaystyle\int{\frac{6 an 3x}{1+\cos 6x}\mathrm{d}x}$. [3] It is given that ${{I}_{n}}=\di

Question

Find $\displaystyle\int{\frac{6	an 3x}{1+\cos 6x}\mathrm{d}x}$. [3] It is given that ${{I}_{n}}=\displaystyle\int_{1}^{\mathrm{e}}{{{\left( \ln x ight)}^{n}}\mathrm{d}x}$, for $n\in {{\mathbb{Z}}^{+}}$. Use integration by parts to show that ${{I}_{n}}=\mathrm{e}-n{{I}_{n-1}}$. [2] The region bounded by the curve $y={{\left( \ln x ight)}^{2}}$, 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. [3]

Accepted Answer

(a) $-\frac12\sec^2 3x+C$; (b)(i) $I_n=e-nI_{n-1}$; volume $\pi(9e-24)$

2021 HCI P1 Q4

2021 HCI P1 Q4

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