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Category: Differentiation

If-cosec-1-x-cot-1-y-show-that-d-2-y-dx-2-1-y-3-0-

Question Number 117482 by Don08q last updated on 12/Oct/20 $$\:\mathrm{If}\:\:\mathrm{cosec}^{−\mathrm{1}} {x}\:\:=\:\:\mathrm{cot}^{−\mathrm{1}} {y},\:\mathrm{show}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:\:+\:\:\frac{\mathrm{1}}{{y}^{\mathrm{3}} }\:\:=\:\:\mathrm{0} \\ $$ Answered by TANMAY PANACEA last updated…

advanced-calculus-prove-that-0-1-ln-x-cos-2-pix-dx-ln-2pi-4-1-8-m-n-1970-

Question Number 117280 by mnjuly1970 last updated on 10/Oct/20 $$\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:…\:{advanced}\:\:{calculus}…\: \\ $$$$ \\ $$$$\:\:\:\:\:\:{prove}\:\:{that}:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\left(\Gamma\left({x}\right)\right).{cos}^{\mathrm{2}} \left(\pi{x}\right){dx} \\ $$$$…

nice-mathematics-proof-1-pi-0-tan-1-1-x-2-dx-ln-2-m-n-1970-

Question Number 117223 by mnjuly1970 last updated on 10/Oct/20 $$\:\:\:\:\:\:\:\:\:…\:{nice}\:\:{mathematics}… \\ $$$$\:\:{proof}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega\:=\:\frac{\mathrm{1}}{\pi}\int_{\mathrm{0}\:} ^{\:\infty} \left[{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{{x}}\right)\right]^{\mathrm{2}} {dx}={ln}\left(\mathrm{2}\right) \\ $$$$\:\:\:\:…\:\:{m}.{n}.\mathrm{1970}… \\ $$ Answered by mathmax…

nice-mathematics-evaluate-lim-n-k-1-n-sin-kpi-4n-1-n-m-n-1970-

Question Number 117086 by mnjuly1970 last updated on 09/Oct/20 $$\:\:\:\:\:\:\:…{nice}\:\:{mathematics}… \\ $$$$\:{evaluate}\:… \\ $$$$\:\:\:\:\:\:\:{lim}_{{n}\rightarrow\infty} \left(\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}{sin}\left(\frac{{k}\pi}{\mathrm{4}{n}}\right)\right)^{\frac{\mathrm{1}}{{n}}} =? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{m}.{n}.\mathrm{1970}… \\ $$$$ \\…