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Category: Integration

let-f-x-e-x-cosx-developp-f-at-fourier-serie-1-find-the-value-of-n-1-n-1-n-2-2-calculate-n-0-1-n-2-1-

Question Number 38209 by prof Abdo imad last updated on 22/Jun/18 $${let}\:{f}\left({x}\right)={e}^{−{x}} {cosx} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\sum_{{n}=−\infty} ^{+\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{1}+{n}^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}}…

cos-2-ln-tan-x-2-tan-x-2-dx-

Question Number 169268 by amin96 last updated on 27/Apr/22 $$\boldsymbol{\Omega}=\int\frac{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{tan}}\frac{\boldsymbol{\mathrm{x}}}{\mathrm{2}}\right)\right)}{\boldsymbol{\mathrm{tan}}\left(\frac{\boldsymbol{\mathrm{x}}}{\mathrm{2}}\right)}\boldsymbol{\mathrm{dx}}=? \\ $$ Answered by Mathspace last updated on 28/Apr/22 $${ln}\left({tan}\left(\frac{{x}}{\mathrm{2}}\right)\right)={t}\:\Rightarrow{tan}\left(\frac{{x}}{\mathrm{2}}\right)={e}^{{t}} \\ $$$$\Rightarrow{x}=\mathrm{2}{arctan}\left({e}^{{t}} \right)\:\Rightarrow \\…