Evaluate exactly $\displaystyle\int_{0}^{\pi }{\cos \left( \frac{x}{2} \right)\cos \left( \frac{x}{3

Question

Evaluate exactly $\displaystyle\int_{0}^{\pi }{\cos \left( \frac{x}{2} \right)\cos \left( \frac{x}{3} \right)+\pi \,\mathrm{d}x}$.[2] The curve with equation $y=2{{\cos }^{2}}x+1$, for $0\le x\le \pi $, is shown in the diagram. Rectangles, each of width $\frac{\pi }{2n}$, are drawn under the curve. Show that the total area ${{S}_{n}}$ of all $2n$ rectangles is given by $\frac{2\pi }{n}\left\{ {{\cos }^{2}}\left( \frac{\pi }{2n} \right)+{{\cos }^{2}}\left( \frac{2\pi }{2n} \right)+...+{{\cos }^{2}}\left( \frac{\left( n-1 \right)\pi }{2n} \right) \right\}+\pi $.[3] Find the exact sum to infinity ${{S}_{\infty }}$.[3]

Accepted Answer

(a) ${\frac{9}{5}+\pi^2}$ (b) (ii) $2\pi$

2020 NJC P1 Q7

2020 NJC P1 Q7

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