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Category: Integration

sin-4x-4sin-4-x-4sin-2-x-1-dx-

Question Number 167438 by cortano1 last updated on 16/Mar/22 $$\:\:\int\:\frac{\mathrm{sin}\:\mathrm{4x}}{\mathrm{4sin}\:^{\mathrm{4}} \mathrm{x}−\mathrm{4sin}\:^{\mathrm{2}} \mathrm{x}+\mathrm{1}}\:\mathrm{dx}=? \\ $$ Answered by MJS_new last updated on 16/Mar/22 $$\frac{\mathrm{sin}\:\mathrm{4}{x}}{\mathrm{1}−\mathrm{4sin}^{\mathrm{2}} \:{x}\:+\mathrm{4sin}^{\mathrm{4}} \:{x}}=\frac{\mathrm{2sin}\:\mathrm{4}{x}}{\mathrm{1}+\mathrm{cos}\:\mathrm{4}{x}}=\frac{\mathrm{2sin}\:\mathrm{2}{x}\:\mathrm{cos}\:\mathrm{2}{x}}{\mathrm{cos}^{\mathrm{2}} \:\mathrm{2}{x}}=…

calculate-1-n-1-H-n-n-2-2-0-1-ln-2-x-Li-2-x-x-dx-

Question Number 167433 by mnjuly1970 last updated on 16/Mar/22 $$ \\ $$$$\:\:\:\:\:\:\:{calculate}\: \\ $$$$\:\:\:\:\mathrm{1}\::\:\:\Omega=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\:{H}_{\:{n}} }{{n}}\:\right)^{\:\mathrm{2}} =\:? \\ $$$$\:\:\:\:\mathrm{2}\::\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{ln}^{\:\mathrm{2}} \left({x}\right).\mathrm{L}{i}_{\:\mathrm{2}} \left({x}\right)}{{x}}\:{dx}\:=\:? \\…

0-pi-x-sin-x-1-cos-2-x-dx-0-pi-1-1-cos-3-x-sin-3-x-dx-0-pi-2-x-cos-x-sin-x-tan-2-x-cot-2-x-dx-0-1-tan-1-1-x-2-dx-

Question Number 167416 by Coronavirus last updated on 16/Mar/22 $$\int_{\:\mathrm{0}} ^{\:\pi} \:\frac{{x}.\mathrm{sin}\:\left({x}\right)}{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \left({x}\right)\:}{dx} \\ $$$$ \\ $$$$\int_{\mathrm{0}} ^{\:\:\pi} \frac{\mathrm{1}}{\mathrm{1}+\mathrm{cos}^{\mathrm{3}} \left({x}\right)\:+\mathrm{sin}^{\mathrm{3}} \left({x}\right)\:}{dx} \\ $$$$ \\ $$$$\int_{\mathrm{0}}…