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Question Number 177906 by cortano1 last updated on 11/Oct/22

$$\underset{0}{\stackrel{1}{\int }}\frac{\sin \mathrm{x}(\cos {}^2\mathrm{x}-\cos {}^2\frac{\pi }{5})(\cos {}^2\mathrm{x}-\cos {}^2\frac{2\pi }{5})}{\sin 5\mathrm{x}}\mathrm{dx}=?$$

Answered by Frix last updated on 11/Oct/22

$$\frac{\sin \mathrm{x}(\cos {}^2\mathrm{x}-\cos {}^2\frac{\pi }{5})(\cos {}^2\mathrm{x}-\cos {}^2\frac{2\pi }{5})}{\sin 5\mathrm{x}}=\frac{1}{16}$$$$\underset{0}{\stackrel{1}{\int }}\frac{dx}{16}=\frac{1}{16}$$

Commented by Frix last updated on 11/Oct/22

$$\sin x=s$$$$\Rightarrow$$$${\cos }^{2}x-{\cos }^2\frac{\pi }{5}=-s^2+\frac{5}{8}-\frac{\sqrt{5}}{8}$$$${\cos }^{2}x-{\cos }^2\frac{2\pi }{5}=-s^2+\frac{5}{8}+\frac{\sqrt{5}}{8}$$$$\sin 5x=16s^5-20s^3+5s$$$$\mathrm{the}\mathrm{rest}\mathrm{is}\mathrm{easy}$$

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