Question Number 92397 by john santu last updated on 06/May/20 $$\int\:\frac{\mathrm{1}}{{x}−\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:{dx}\: \\ $$$$\left[\:{x}\:=\:\mathrm{sin}\:{w}\:\right]\: \\ $$$$\int\:\frac{\mathrm{cos}\:\mathrm{w}\:\mathrm{dw}}{\mathrm{sin}\:\mathrm{w}−\mathrm{cos}\:\mathrm{w}}\:=\:\int\:\frac{\mathrm{dw}}{\mathrm{tan}\:\mathrm{w}−\mathrm{1}} \\ $$$$=\:\int\:\frac{\mathrm{sec}^{\mathrm{2}} \:\mathrm{w}\:\mathrm{dw}}{\left(\mathrm{tan}\:\mathrm{w}−\mathrm{1}\right)\mathrm{sec}^{\mathrm{2}} \:\mathrm{w}} \\ $$$$=\:\int\:\frac{\mathrm{du}}{\left(\mathrm{u}−\mathrm{1}\right)\left(\mathrm{u}^{\mathrm{2}} +\mathrm{1}\right)}\:;\:\left[\:\mathrm{u}\:=\:\mathrm{tan}\:\mathrm{w}\:\right]\: \\ $$$$=\:\int\:\frac{\mathrm{du}}{\mathrm{2}\left(\mathrm{u}−\mathrm{1}\right)}−\int\:\frac{\mathrm{u}\:\mathrm{du}\:}{\mathrm{2}\left(\mathrm{u}^{\mathrm{2}}…
Question Number 92394 by john santu last updated on 06/May/20 $$\int\:\mathrm{ln}\:\left(\sqrt{\mathrm{1}−{x}}\:+\:\sqrt{\mathrm{1}+{x}}\:\right)\:{dx}\: \\ $$ Commented by mathmax by abdo last updated on 06/May/20 $${I}\:=\int{ln}\left(\sqrt{\mathrm{1}−{x}}+\sqrt{\mathrm{1}+{x}}\right){dx}\:\:{by}\:{parts}\: \\ $$$${I}\:={x}\left(\sqrt{\mathrm{1}−{x}}+\sqrt{\mathrm{1}+{x}}\right)−\int\:{x}\frac{\left(\sqrt{\mathrm{1}−{x}}+\sqrt{\mathrm{1}+{x}}\right)^{'}…
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Question Number 157870 by Odhiambojr last updated on 29/Oct/21 $${find}\:{the}\:{integral}: \\ $$$$\int\left\{\left(\mathrm{3}{x}+\mathrm{1}\right)/\left({x}^{\mathrm{2}} +\mathrm{4}\right)\right\}{dx} \\ $$ Answered by puissant last updated on 29/Oct/21 $$\Omega=\int\:\frac{\mathrm{3}{x}+\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{4}}{dx}\:=\:\int\frac{\mathrm{3}{x}}{{x}^{\mathrm{2}} +\mathrm{4}}{dx}+\int\frac{{dx}}{{x}^{\mathrm{2}}…
Question Number 92323 by Power last updated on 06/May/20 Commented by Prithwish Sen 1 last updated on 06/May/20 $$\mathrm{split}\:\boldsymbol{\mathrm{x}}+\mathrm{3}\:\boldsymbol{\mathrm{into}}\:\frac{\mathrm{1}}{\mathrm{8}}\left(\mathrm{8}\boldsymbol{\mathrm{x}}+\mathrm{4}\right)+\frac{\mathrm{5}}{\mathrm{2}} \\ $$ Commented by Power last…
Question Number 26781 by shubhabrata04@gmail.com last updated on 29/Dec/17 Commented by prakash jain last updated on 29/Dec/17 $$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{log}\:\left(\mathrm{1}−{x}\right)−\mathrm{log}\:{x}\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{log}\:\left(\mathrm{1}−{x}\right){dx}−\int_{\mathrm{0}} ^{\mathrm{1}}…
Question Number 26758 by abdo imad last updated on 28/Dec/17 $${let}\:{give}\:{D}=\left\{\left(\:\:{x},{y}\:\right)\in\mathbb{R}^{\mathrm{2}} /{x}^{\mathrm{2}} −{x}\:+{y}^{\mathrm{2}} \leqslant\:\mathrm{4}\:{and}\:\:\mathrm{0}\leqslant{y}\leqslant\mathrm{1}\right\} \\ $$$${calculate}\:\int\int_{{D}} {ln}\left({xy}\right)\sqrt{\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} {dxdy}\:\:} \\ $$ Commented by abdo imad…
Question Number 26759 by abdo imad last updated on 29/Dec/17 $${find}\:\:{the}\:{value}\:{of}\:\:\:\int_{\mathrm{0}} ^{\:\propto} \:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left({x}+\mathrm{3}\right)}\:. \\ $$ Commented by abdo imad last updated on 30/Dec/17 $${we}\:{use}\:{the}\:{changement}\:{x}+\mathrm{2}={t} \\…
Question Number 26755 by abdo imad last updated on 28/Dec/17 $${find}\:\:\int\:\:\frac{{dx}}{{x}^{\mathrm{6}} −\mathrm{1}}\:\:. \\ $$ Answered by $@ty@m last updated on 29/Dec/17 $$\int\frac{{dx}}{\left({x}^{\mathrm{3}} +\mathrm{1}\right)\left({x}^{\mathrm{3}} −\mathrm{1}\right)} \\…