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Category: Integration

nice-calculus-evaluate-n-1-1-n-n-2-1-sin-n-n-2-

Question Number 129651 by mnjuly1970 last updated on 17/Jan/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:…\:{nice}\:\:\:{calculus}… \\ $$$$\:\:\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} +\mathrm{1}}\:\left(\frac{{sin}\left({n}\right)}{{n}}\right)^{\mathrm{2}} =? \\ $$$$ \\ $$ Terms of…

dx-1-x-12-

Question Number 129645 by liberty last updated on 17/Jan/21 $$\:\:\:\:\digamma\:=\:\int\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{12}} }\:. \\ $$ Answered by mathmax by abdo last updated on 17/Jan/21 $$\mathrm{roots}\:\mathrm{of}\:\mathrm{z}^{\mathrm{12}} +\mathrm{1}=\mathrm{0}\:\:\:\Rightarrow\mathrm{z}^{\mathrm{12}} \:=\mathrm{e}^{\mathrm{i}\pi\:+\mathrm{2ik}\pi}…

dx-x-2-1-2-k-1-n-1-k-k-1-2k-1-

Question Number 129635 by pticantor last updated on 17/Jan/21 $$\:\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\int\frac{\boldsymbol{{dx}}}{\left(\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }=? \\ $$$$\underset{\boldsymbol{{k}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\frac{\mathrm{1}}{\boldsymbol{{k}}\left(\boldsymbol{{k}}+\mathrm{1}\right)\left(\mathrm{2}\boldsymbol{{k}}+\mathrm{1}\right)}=? \\ $$ Commented by liberty…