Use the substitution $\tan y={{\mathrm{e}}^{x}}$ to find \(\displaystyle\displaystyle \displaystyl

Question

Use the substitution $\tan y={{\mathrm{e}}^{x}}$ to find $\displaystyle\displaystyle \displaystyle\int{\frac{1}{{{\mathrm{e}}^{x}}(1+{{\mathrm{e}}^{2x}})}\mathrm{d}x}$.[4] The region bounded by the $x$-axis, the curve $y={{\mathrm{e}}^{{{x}^{2}}+1}}$, the lines $x=0$ and $x=1$ is rotated about the $y$-axis to form a solid of revolution of volume $V$ . Find the exact value of $V$.[6]

Accepted Answer

(a) $-e^{-x}-\tan^{-1}(e^x)+C$ (b) $\pi e(e-1)$

2012 RVHS P2 Q2

2012 RVHS P2 Q2

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