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Category: Differentiation

what-is-maximum-value-of-y-3x-4-2-sin-2x-

Question Number 77574 by jagoll last updated on 08/Jan/20 $${what}\:{is}\:{maximum}\: \\ $$$${value}\:{of}\:{y}=\:\left(\mathrm{3}{x}+\mathrm{4}\right)^{\mathrm{2}} ×\mathrm{sin}\:\left(\mathrm{2}{x}\right) \\ $$ Commented by MJS last updated on 08/Jan/20 $$\mathrm{sin}\:\mathrm{2}{x}\:=−\mathrm{1}\:\Rightarrow\:{x}={n}\pi−\frac{\pi}{\mathrm{4}}=\frac{\mathrm{4}{n}−\mathrm{1}}{\mathrm{4}}\pi \\ $$$$\mathrm{sin}\:\mathrm{2}{x}\:=+\mathrm{1}\:\Rightarrow\:{x}={n}\pi+\frac{\pi}{\mathrm{4}}=\frac{\mathrm{4}{n}+\mathrm{1}}{\mathrm{4}}\pi…

Evaluate-0-pi-4-ln-tan-x-sin-pi-e-2x-sin-pi-e-x-cos-pi-e-x-2-dx-

Question Number 143086 by mnjuly1970 last updated on 09/Jun/21 $$ \\ $$$$\:\:{Evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\Omega:=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \frac{{ln}\left({tan}\left({x}\right)\right).{sin}^{\pi^{{e}} } \left(\mathrm{2}{x}\right)}{\left({sin}^{\pi^{{e}} } \left({x}\right)+{cos}^{\pi^{{e}} } \left({x}\right)\right)^{\mathrm{2}} }{dx} \\ $$$$…

Question-142935

Question Number 142935 by mnjuly1970 last updated on 07/Jun/21 Answered by qaz last updated on 07/Jun/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}^{\mathrm{2}} \left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}}\mathrm{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}^{\mathrm{2}}…