Question Number 212139 by mnjuly1970 last updated on 03/Oct/24 $$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\mathrm{I}_{{n}} \:=\:\int_{−\pi} ^{\:\pi} \frac{\:\mathrm{sin}\left({nx}\:\right)}{\left(\mathrm{1}\:+\:{e}^{{x}} \right)\mathrm{sin}{x}}\:{dx}\:=?\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\: \\ $$ Answered…
Question Number 212099 by vahid last updated on 30/Sep/24 Answered by mehdee7396 last updated on 30/Sep/24 $$\int\frac{\mathrm{1}+{tan}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{\mathrm{1}−{tan}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{dx}\:\:\:\:\:;\:{let}\:\:\:{tan}\frac{{x}}{\mathrm{2}}={u} \\ $$$$=\int\frac{\mathrm{2}{u}}{\mathrm{1}−{u}^{\mathrm{2}} }{du}={ln}\frac{\mathrm{1}+{u}}{\mathrm{1}−{u}}+{c} \\ $$$$={ln}\left({tan}\left(\frac{\pi}{\mathrm{4}}+\frac{{x}}{\mathrm{2}}\right)\right)+{c}\:\:\checkmark \\…
Question Number 212053 by universe last updated on 28/Sep/24 $$ \\ $$$$\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\int_{\mathrm{0}} ^{\:\boldsymbol{{y}}} \:\boldsymbol{{e}}^{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} } \boldsymbol{{dx}}\right)\boldsymbol{{dy}}\:+\int_{\mathrm{1}} ^{\mathrm{2}} \left(\int_{\mathrm{0}} ^{\:\mathrm{2}−\boldsymbol{{y}}} \:\boldsymbol{{e}}^{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} }…
Question Number 212023 by Spillover last updated on 27/Sep/24 Answered by Frix last updated on 27/Sep/24 $$\int\:\frac{\left({x}+\mathrm{1}\right)\mathrm{tan}\:{x}}{\left(\mathrm{1}+\mathrm{tan}\:{x}\right)^{\mathrm{2}} }{dx}= \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\left({x}+\mathrm{1}−\frac{\mathrm{1}}{\underset{\left[{t}=\mathrm{tan}\:{x}\right]} {\underbrace{\mathrm{1}+\mathrm{sin}\:\mathrm{2}{x}}}}−\frac{{x}}{\underset{\left[\mathrm{by}\:\mathrm{parts}\right]} {\underbrace{\mathrm{1}+\mathrm{sin}\:\mathrm{2}{x}}}}\right){dx}= \\ $$$$… \\…
Question Number 211995 by SVMEHTA last updated on 26/Sep/24 $$\int\sqrt{}{tanx}\:{dx} \\ $$ Commented by Frix last updated on 26/Sep/24 $$\mathrm{Use}\:{t}=\sqrt{\mathrm{tan}\:{x}} \\ $$ Commented by BHOOPENDRA…
Question Number 212007 by Nadirhashim last updated on 26/Sep/24 $$\:\:\:\:\int\boldsymbol{{sin}}\left(\boldsymbol{{x}}\right)\:\sqrt[{\mathrm{3}}]{\boldsymbol{{tan}}\left(\boldsymbol{{x}}\right)}\:.\boldsymbol{{dx}} \\ $$ Answered by Ghisom last updated on 26/Sep/24 $$=\int\left(\mathrm{cos}\:{x}\right)^{−\mathrm{1}/\mathrm{3}} \left(\mathrm{sin}\:{x}\right)^{\mathrm{4}/\mathrm{3}} {dx}= \\ $$$$=\frac{\mathrm{3}}{\mathrm{7}}\:_{\mathrm{2}} {F}_{\mathrm{1}}…
Question Number 212001 by mnjuly1970 last updated on 26/Sep/24 $$ \\ $$$$\:\:\:\:{prove}\:\:{that}: \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\underset{{k}\in\mathbb{Z}} {\sum}\:\frac{\:\left(−\mathrm{1}\right)^{{k}} }{\:{x}\:+\:{k}\pi}\:=\:\frac{\mathrm{1}}{{sin}\left({x}\right)}\:\:\: \\ $$$$\:\:\:\:\:\:\:\:−−−−−−−−− \\ $$ Answered…
Question Number 211857 by Frix last updated on 22/Sep/24 $$\underset{\mathrm{1}} {\overset{\infty} {\int}}\left(\frac{\mathrm{tan}^{−\mathrm{1}} \:{x}}{{x}}×\frac{\mathrm{ln}\:{x}}{{x}}\right){dx}=? \\ $$ Answered by Berbere last updated on 23/Sep/24 $$\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{tan}^{−\mathrm{1}}…
Question Number 211778 by Mr.D.N. last updated on 20/Sep/24 $$\:\:\mathrm{Show}\:\mathrm{that}: \\ $$$$\:\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{4}}} \:\boldsymbol{\mathrm{sin}}^{\mathrm{4}} \boldsymbol{\mathrm{x}}\:\mathrm{2}\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{dx}}\:=\:\frac{\mathrm{3}\pi\:−\mathrm{4}}{\mathrm{192}} \\ $$ Commented by BHOOPENDRA last updated on 20/Sep/24 $${this}\:{result}\:{can}\:{to}\:{possible}…
Question Number 211749 by universe last updated on 19/Sep/24 $${volume}\:{bounded}\:{by}\:{the}\:{curve} \\ $$$$\:{z}\:=\:\sqrt{\mathrm{3}{x}^{\mathrm{2}} +\mathrm{3}{y}^{\mathrm{2}} }\:\:\:{and}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} =\:\mathrm{6}^{\mathrm{2}} \\ $$ Answered by mr W last updated…