Use the integration by parts to show that \(\displaystyle\int{{{e}^{x}}\sin x\,\mathrm{d}x=\fra

Question

Use the integration by parts to show that $\displaystyle\int{{{e}^{x}}\sin x\,\mathrm{d}x=\frac{{{e}^{x}}}{2}(\sin x-\cos x)+c}$, where $c$ is an arbitrary constant. The diagram above shows the graphs of a curve $C$ given by $y={{e}^{\frac{x}{2}}}\sqrt{\sin x}-{{e}^{\frac{\pi }{4}}}$ for $0\le x\le \pi $. The region $R$ is bounded by the $x$-axis and the part of $C$ between $x=0$ and $x=\frac{5}{2}$. Find, correct to $3$ significant figures, the area of region $R$. The region $S$ is bounded by $C$ and the line $y=-{{e}^{\frac{\pi }{4}}}$. Write down the equation of the curve obtained when $C$ is translated ${{e}^{\frac{\pi }{4}}}$ units in the positive $y$-direction. Hence, show that the volume $V$ of the solid formed when $S$ is rotated $2\pi $ radians about the line $y=-{{e}^{\frac{\pi }{4}}}$ is given by $V=a({{e}^{\pi }}+b)$, where $a$ and $b$ are exact constants to be determined.

Accepted Answer

(i) $\frac{\mathrm e^x}{2}(\sin x-\cos x)+c$ (shown). (ii)(a) $1.85$ square units. (ii)(b) Translated curve $y=\mathrm e^{\frac{x}{2}}\sqrt{\sin x}$, $0\le x\le\pi$; $V=\frac\pi2(\mathrm e^\pi+1)$, $a=\frac\pi2$, $b=1$ (shown).

2013 NJC P1 Q12

2013 NJC P1 Q12

Solution:

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