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Author: Tinku Tara

Question-8653

Question Number 8653 by tawakalitu last updated on 19/Oct/16 Answered by prakash jain last updated on 19/Oct/16 $$\mathrm{2sin}\:{x}+\mathrm{3cos}\:{y}=\mathrm{5} \\ $$$$\mathrm{sin}\:{x}\leqslant\mathrm{1} \\ $$$$\mathrm{cos}\:{y}\leqslant\mathrm{1} \\ $$$$\mathrm{If}\:\mathrm{2sin}\:{x}+\mathrm{3cos}\:{y}=\mathrm{5} \\…

Prove-1-2-n-1-a-n-cos-nx-1-a-2-1-2acos-x-a-2-a-lt-1-And-calculate-0-pi-x-2-1-8sin-2-x-dx-5pi-3-36-pi-6-ln-2-2-

Question Number 139708 by qaz last updated on 30/Apr/21 $${Prove}:\:\:\:\mathrm{1}+\mathrm{2}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{a}^{{n}} \mathrm{cos}\:\left({nx}\right)=\frac{\mathrm{1}−{a}^{\mathrm{2}} }{\mathrm{1}−\mathrm{2}{a}\mathrm{cos}\:{x}+{a}^{\mathrm{2}} },\:\:\:\:\:\:\left(\mid{a}\mid<\mathrm{1}\right) \\ $$$${And}\:{calculate}::\:\:\int_{\mathrm{0}} ^{\pi} \frac{{x}^{\mathrm{2}} }{\mathrm{1}+\mathrm{8sin}\:^{\mathrm{2}} {x}}{dx}=\frac{\mathrm{5}\pi^{\mathrm{3}} }{\mathrm{36}}−\frac{\pi}{\mathrm{6}}{ln}^{\mathrm{2}} \mathrm{2} \\ $$…