Question Number 85701 by jagoll last updated on 24/Mar/20 $$\int\:\frac{\sqrt{\mathrm{3x}−\mathrm{1}}}{\:\sqrt{\mathrm{2x}+\mathrm{1}}}\:\mathrm{dx}\: \\ $$ Commented by john santu last updated on 24/Mar/20 $${let}\:{t}\:=\:\sqrt{\mathrm{2}{x}+\mathrm{1}}\:\Rightarrow\:{x}\:=\:\frac{{t}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2}} \\ $$$$\int\:\frac{\sqrt{\frac{\mathrm{3}{t}^{\mathrm{2}} −\mathrm{3}}{\mathrm{2}}−\mathrm{1}}}{{t}}×{t}\:{dt}\:=\:…
Question Number 20162 by ajfour last updated on 23/Aug/17 $${Compute}\:{the}\:{volume}\:{of}\:{a}\:{solid} \\ $$$${bounded}\:{by}\:{a}\:{surface}\:{with}\:{equation} \\ $$$$\:\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right)^{\mathrm{2}} ={a}^{\mathrm{3}} {x}\:. \\ $$ Answered by ajfour last…
Question Number 151230 by mnjuly1970 last updated on 19/Aug/21 $$ \\ $$$$\:\:\:\:{prove}: \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{ln}\:\left(\:\mathrm{1}+{x}^{\:\mathrm{2}} \right)}{{x}^{\:\mathrm{2}} \left(\mathrm{1}+{x}^{\:\mathrm{2}} \right)}{dx}=\:\pi\:{ln}\left(\frac{{e}}{\mathrm{2}}\:\right)\:.. \\ $$ Commented by Lordose last…
Question Number 20157 by ajfour last updated on 23/Aug/17 Commented by ajfour last updated on 23/Aug/17 $${Compute}\:{the}\:{volume}\:{common}\:{to}\:{the} \\ $$$${sphere}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} =\mathrm{4}\:\:{and}\:\:{the}\:{inside}\:{of} \\ $$$${the}\:{paraboloid}\:\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}}…
Question Number 151220 by peter frank last updated on 19/Aug/21 $$\mathrm{show}\:\mathrm{that}\: \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{cos}\:\mathrm{x}}{\mathrm{cos}\:\mathrm{x}+\mathrm{sin}\:\mathrm{x}+\mathrm{1}}\mathrm{dx}=\frac{\mathrm{1}}{\mathrm{4}}\left(\pi−\mathrm{2ln}\:\mathrm{2}\right) \\ $$ Answered by ArielVyny last updated on 19/Aug/21 $${t}={tan}\left(\frac{{x}}{\mathrm{2}}\right)\rightarrow{dt}=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+{t}^{\mathrm{2}}…
Question Number 151221 by peter frank last updated on 19/Aug/21 $$\int\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{sin}\:\left(\mathrm{x}−\mathrm{a}\right)}\mathrm{dx} \\ $$ Commented by puissant last updated on 19/Aug/21 $${Q}=\int\frac{{sinx}}{{sin}\left({x}−{a}\right)}{dx} \\ $$$$=\int\frac{{sin}\left({x}−{a}+{a}\right)}{{sin}\left({x}−{a}\right)}{dx}\:\: \\ $$$$=\int\frac{{sin}\left({x}−{a}\right){cosa}+{cos}\left({x}−{a}\right){sina}}{{sin}\left({x}−{a}\right)}{dx}…
Question Number 85677 by john santu last updated on 24/Mar/20 $$\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\mathrm{sin}\:\left(\frac{\mathrm{21}{x}}{\mathrm{2}}\right)}{\mathrm{sin}\:\left(\frac{{x}}{\mathrm{2}}\right)}\:{dx}\: \\ $$ Commented by jagoll last updated on 24/Mar/20 $$\mathrm{i}\:\mathrm{don}'\mathrm{t}\:\mathrm{have}\:\mathrm{a}\:\mathrm{simple}\:\mathrm{way}\:\mathrm{to}\:\mathrm{solve} \\ $$$$\mathrm{it}…
Question Number 85669 by john santu last updated on 23/Mar/20 $$\int\:\frac{\sqrt{\mathrm{1}+{x}}}{\:\sqrt{\mathrm{1}−{x}}}\:{dx} \\ $$$$ \\ $$ Commented by mathmax by abdo last updated on 24/Mar/20 $${I}\:=\int\:\frac{\sqrt{\mathrm{1}+{x}}}{\:\sqrt{\mathrm{1}−{x}}}{dx}\:\Rightarrow\:{I}\:=\int\sqrt{\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}}{dx}\:{we}\:{use}\:{the}\:{changement}\:…
Question Number 85667 by john santu last updated on 23/Mar/20 $$\int\:\frac{{dx}}{\:\sqrt{\mathrm{1}−\mathrm{sin}\:\mathrm{2}{x}}}\: \\ $$ Answered by som(math1967) last updated on 24/Mar/20 $$\int\frac{{dx}}{\left({cosx}−{sinx}\right)} \\ $$$$\int\frac{{dx}}{\:\sqrt{\mathrm{2}}{cos}\left({x}+\frac{\pi}{\mathrm{4}}\right)} \\ $$$$\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\int{sec}\left({x}+\frac{\pi}{\mathrm{4}}\right){dx}…
Question Number 85648 by sakeefhasan05@gmail.com last updated on 23/Mar/20 $$\int\frac{\left(\mathrm{x}^{\mathrm{3}} −\mathrm{4}\right)}{\left(\mathrm{x}+\mathrm{1}\right)}\mathrm{dx} \\ $$ Answered by petrochengula last updated on 23/Mar/20 $$=\int\frac{{x}^{\mathrm{3}} +\mathrm{1}−\mathrm{5}}{\left({x}+\mathrm{1}\right)}{dx} \\ $$$$=\int\frac{{x}^{\mathrm{3}} +\mathrm{1}}{{x}+\mathrm{1}}{dx}−\mathrm{5}\int\frac{\mathrm{1}}{{x}+\mathrm{1}}{dx}…