Question Number 211708 by Skyneless last updated on 17/Sep/24 Answered by Frix last updated on 18/Sep/24 $$\int\sqrt{\mathrm{tan}\:{x}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\mathrm{2tan}^{−\mathrm{1}} \:\sqrt{\mathrm{tan}\:{x}}\right] \\ $$$$=\int\frac{\mathrm{1}−\mathrm{cos}\:{t}}{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \:{t}}{dt} \\ $$$$…
Question Number 211696 by mnjuly1970 last updated on 16/Sep/24 $$ \\ $$$$\:\:\:\:\:\underset{\mathrm{0}} {\int}^{\:\mathrm{1}} \frac{\:\:\mathrm{1}}{\left(\:\mathrm{2}\:+\mathrm{2}{x}\:+\:{x}^{\mathrm{2}} \:\right)^{\mathrm{3}} }\:{dx}=\:? \\ $$$$\:\:\:\:\:\:\underbrace{\underset{\:\:\:\:\overset{\mathrm{Improper}\:\mathrm{integral}\:} {\:}\:\:\:\:\:} {\:}} \\ $$$$\:\:\:\:\:\:\:\:−−−−−−−−− \\ $$ Answered…
Question Number 211528 by mnjuly1970 last updated on 12/Sep/24 $$ \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{tanh}^{−\mathrm{1}} \:\left({x}^{\mathrm{2}} \:\right)}{{x}^{\:\mathrm{2}} }\:{dx}=\:?\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$…
Question Number 211486 by mnjuly1970 last updated on 10/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:{Mathematical}\:\:\:\:\:{Analysis}… \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:{f}:\mathbb{R}\:\rightarrow\:\mathbb{R}\:{is}\:{diffrentiable}\:{function}, \\ $$$$\:\:\:\:\:{f}\:\:{and}\:\:{f}\:'\:,\:{has}\:{no}\:{common}\:{zero} \\ $$$$\:\:\:\:\:{on}\:\:\mathbb{R}\:. \\ $$$$\:\:\:\:\:\:{prove}\:{that}\:{the}\:{following}\:{set}\:{is}\:{finite}. \\ $$$$ \\…
Question Number 211421 by universe last updated on 08/Sep/24 $$\:\:\:\:\mathrm{if}\:\mathrm{a}_{\mathrm{n}} \:=\:\mathrm{n}^{\mathrm{4}} \int_{\mathrm{n}} ^{\mathrm{n}+\mathrm{1}} \:\frac{\mathrm{x}\:\mathrm{dx}}{\mathrm{1}+\mathrm{x}^{\mathrm{5}} }\:\:\mathrm{then} \\ $$$$\:\:\:\:\left(\mathrm{1}\right)\:\Sigma\mathrm{a}_{\mathrm{n}} \:\mathrm{is}\:\mathrm{convergent}\:\mathrm{or}\:\mathrm{divergent}?? \\ $$$$\:\:\:\:\left(\mathrm{2}\right)\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{a}_{\mathrm{n}\:} \:=\:?? \\ $$ Answered…
Question Number 211400 by Spillover last updated on 08/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{\mathrm{3}} −\mathrm{3}{x}^{\mathrm{2}} +\mathrm{3}{x}−\mathrm{1}}{{x}^{\mathrm{4}} +\mathrm{4}{x}^{\mathrm{3}} +\mathrm{6}{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{1}}{dx} \\ $$$$ \\ $$ Answered by…
Question Number 211399 by Spillover last updated on 08/Sep/24 $$\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{4}} \right)\sqrt{\mathrm{1}+{x}^{\mathrm{4}} −{x}^{\mathrm{2}} }} \\ $$$$ \\ $$ Commented by Frix last updated on…
Question Number 211344 by Nadirhashim last updated on 06/Sep/24 $$\:\:\:{find}\:\int\frac{\boldsymbol{{dx}}}{\boldsymbol{{sin}}^{\mathrm{3}} \left(\boldsymbol{{x}}\right)\:\boldsymbol{{cos}}^{\mathrm{5}} \left(\boldsymbol{{x}}\right)}\:.\boldsymbol{{dx}}\: \\ $$ Answered by Frix last updated on 06/Sep/24 $$\int\frac{{dx}}{\mathrm{cos}^{\mathrm{5}} \:{x}\:\mathrm{sin}^{\mathrm{3}} \:{x}}=\int\frac{\left(\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \:{x}\right)^{\mathrm{4}}…
Question Number 211245 by Ghisom last updated on 01/Sep/24 $$\mathrm{prove}: \\ $$$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{{t}^{\alpha−\mathrm{1}} }{{t}^{\pi} +\mathrm{1}}{dt}=\frac{\mathrm{1}}{\mathrm{sin}\:\alpha} \\ $$ Answered by Frix last updated on 02/Sep/24…
Question Number 211155 by Spillover last updated on 30/Aug/24 $$\int\frac{\left(\mathrm{sin}\:^{{n}} \left(\theta\right)−\mathrm{sin}\:\left(\theta\right)\right)^{\frac{\mathrm{1}}{{n}}} \mathrm{cos}\:\left(\theta\right)}{\mathrm{sin}\:^{{n}+\mathrm{1}} \left(\theta\right)}{d}\theta \\ $$$$ \\ $$ Answered by Frix last updated on 30/Aug/24 $$=\int\frac{{d}\theta}{\mathrm{sin}\:\theta}\:−\int\left(\mathrm{sin}\:\theta\right)^{\frac{\mathrm{1}−{n}−{n}^{\mathrm{2}}…