Question Number 20385 by tammi last updated on 26/Aug/17 $$\int\sqrt{\mathrm{16}−\mathrm{9}{x}^{\mathrm{2}} {dx}} \\ $$ Answered by ajfour last updated on 26/Aug/17 $$=\mathrm{3}\int\sqrt{\left(\frac{\mathrm{4}}{\mathrm{3}}\right)^{\mathrm{2}} −{x}^{\mathrm{2}} }\:{dx} \\ $$$$=\frac{\mathrm{3}{x}}{\mathrm{2}}\sqrt{\left(\frac{\mathrm{4}}{\mathrm{3}}\right)^{\mathrm{2}}…
Question Number 20386 by tammi last updated on 26/Aug/17 $$\int\sqrt{\mathrm{1}−{a}^{\mathrm{2}} {x}^{\mathrm{2}} {dx}} \\ $$ Answered by $@ty@m last updated on 26/Aug/17 $${Let}\:{x}=\frac{{sin}\theta}{{a}} \\ $$$$\Rightarrow{dx}=\frac{\mathrm{1}}{{a}}{cos}\theta{d}\theta \\…
Question Number 20383 by tammi last updated on 26/Aug/17 $$\int\frac{{dx}}{{x}\sqrt{\mathrm{2}+\left({x}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} }} \\ $$ Answered by $@ty@m last updated on 26/Aug/17 $${let}\:\mathrm{2}+{x}^{\frac{\mathrm{1}}{\mathrm{3}}} ={t}^{\mathrm{2}} \Rightarrow{x}=\left({t}^{\mathrm{2}} −\mathrm{2}\right)^{\mathrm{3}} \\…
Question Number 85914 by frc2crc last updated on 26/Mar/20 $${Let}\:{I}_{{n}} =\overset{\pi/\mathrm{4}} {\int}_{\mathrm{0}\:} \left(\frac{\mathrm{1}−\mathrm{tan}\:{A}}{\mathrm{1}+\mathrm{tan}\:{A}}\right)^{{n}} {dA}\:\:{what}\:{is} \\ $$$${the}\:{Laplace}\:{Transform}\:{and}\:{the} \\ $$$${Fourier}\:{Transform} \\ $$ Answered by mind is power…
Question Number 85909 by jagoll last updated on 26/Mar/20 $$\int\:\mathrm{sin}^{−\mathrm{1}} \:\left(\sqrt{\frac{\mathrm{x}}{\mathrm{a}+\mathrm{x}}}\right)\:\mathrm{dx}\:,\:\mathrm{a}\:>\:\mathrm{0} \\ $$ Commented by john santu last updated on 26/Mar/20 $${let}\:\sqrt{\frac{{x}}{{a}+{x}}}\:=\:{n}\:\Rightarrow\:{x}\:=\:\frac{{an}^{\mathrm{2}} }{\mathrm{1}−{n}^{\mathrm{2}} } \\…
Question Number 151447 by mnjuly1970 last updated on 21/Aug/21 $$ \\ $$$$\:\:\:\:\:\:{prove}\:{that}… \\ $$$$\:\:\:\mathrm{I}:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{ln}\:\left({x}\:\right)}{\mathrm{1}+\:{e}^{\:{x}} }\:{dx}=\:\frac{−\mathrm{1}}{\mathrm{2}}\:{ln}^{\:\mathrm{2}} \left(\mathrm{2}\right)\:..\blacksquare \\ $$ Answered by Lordose last updated…
Question Number 85896 by ar247 last updated on 25/Mar/20 Commented by abdomathmax last updated on 25/Mar/20 $${I}=\int\:\:\frac{\mathrm{5}{x}−\mathrm{5}}{\mathrm{3}{x}^{\mathrm{2}} −\mathrm{8}{x}−\mathrm{3}}{dx} \\ $$$$\mathrm{3}{x}^{\mathrm{2}} −\mathrm{8}{x}−\mathrm{3}=\mathrm{0}\:\rightarrow\Delta^{'} =\mathrm{4}^{\mathrm{2}} −\left(\mathrm{3}\right)\left(−\mathrm{3}\right)\:=\mathrm{16}+\mathrm{9}=\mathrm{25} \\ $$$${x}_{\mathrm{1}}…
Question Number 151425 by peter frank last updated on 21/Aug/21 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{dx}}{\left(\mathrm{cos}\:\mathrm{x}+\sqrt{\mathrm{3}}\:\mathrm{sin}\:\mathrm{x}\right)^{\mathrm{2}} }\mathrm{dx}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}} \\ $$ Answered by Olaf_Thorendsen last updated on 21/Aug/21 $$\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}}…
Question Number 151428 by peter frank last updated on 21/Aug/21 $$\int\frac{\mathrm{x}+\mathrm{2}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{3x}+\mathrm{3}\right)\sqrt{\mathrm{x}+\mathrm{1}}}\mathrm{dx} \\ $$$$ \\ $$ Answered by MJS_new last updated on 21/Aug/21 $$\left(\mathrm{1}\right)\:\mathrm{trying}\:\mathrm{something} \\…
Question Number 151429 by peter frank last updated on 21/Aug/21 $$\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{xdx}}{\left(\mathrm{a}^{\mathrm{2}} \mathrm{cos}\:^{\mathrm{2}} \mathrm{x}+\mathrm{b}^{\mathrm{2}} \mathrm{sin}\:^{\mathrm{2}} \mathrm{x}\right)^{\mathrm{2}} } \\ $$ Answered by Olaf_Thorendsen last updated…