Question Number 116586 by Bird last updated on 05/Oct/20 $$\left.\mathrm{1}\right)\:{explicite}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left(\mathrm{1}+{x}\left(\mathrm{2}+{t}^{\mathrm{2}} \right)\right)}{\mathrm{2}+{t}^{\mathrm{2}} }{dt} \\ $$$${withx}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right){determine}\:{values}\:{of} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left(\mathrm{3}+{t}^{\mathrm{2}} \right)}{\mathrm{2}+{t}^{\mathrm{2}} }{dt}\:{and}\: \\…
Question Number 51047 by Necxx last updated on 23/Dec/18 Commented by Necxx last updated on 23/Dec/18 $${i}\:{have}\:{been}\:{seeing}\:{this}\:{question} \\ $$$${for}\:{a}\:{long}\:{time}\:{now}.\:{Please}\:{help} \\ $$$${if}\:{you}\:{can}. \\ $$$$ \\ $$$${Thanks}\:{in}\:{advance}.…
Question Number 116564 by bemath last updated on 05/Oct/20 $$\int\:\frac{\mathrm{dx}}{\mathrm{x}+\sqrt[{\mathrm{3}\:}]{\mathrm{x}}}\:? \\ $$ Answered by MJS_new last updated on 05/Oct/20 $$\int\frac{{dx}}{{x}^{\mathrm{1}/\mathrm{3}} \left({x}^{\mathrm{2}/\mathrm{3}} +\mathrm{1}\right)}= \\ $$$$\:\:\:\:\:\left[{t}={x}^{\mathrm{2}/\mathrm{3}} +\mathrm{1}\:\rightarrow\:{dx}=\frac{\mathrm{3}}{\mathrm{2}}{x}^{\mathrm{1}/\mathrm{3}}…
Question Number 116560 by Bird last updated on 04/Oct/20 $${if}\:{arctan}\left({x}+{iy}\right)\:={a}+{ib} \\ $$$${with}\:{a}\:{and}\:{b}\:{reals}\:{determine} \\ $$$${a}\:{and}\:{b} \\ $$ Commented by MJS_new last updated on 05/Oct/20 $$\mathrm{arctan}\:\left({x}+\mathrm{i}{y}\right)\:={a}+\mathrm{i}{b} \\…
Question Number 116557 by Bird last updated on 04/Oct/20 $${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left(\mathrm{1}+{x}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\right.}{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$$${with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left(\mathrm{2}+{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$${and}\:\int_{\mathrm{0}} ^{\infty}…
Question Number 116554 by Bird last updated on 04/Oct/20 $${study}\:{the}\:{integral}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{sin}^{{n}} {x}}{{x}^{{n}} }{dx} \\ $$$${n}\:{integr}\:{and}\:{n}\geqslant\mathrm{2} \\ $$ Answered by Olaf last updated on 05/Oct/20…
Question Number 116533 by Bird last updated on 04/Oct/20 $${find}\:{u}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\frac{{sin}^{{n}} {x}}{{x}}{dx}\:\:\:\left({n}\geqslant\mathrm{1}\right) \\ $$ Answered by Olaf last updated on 04/Oct/20 $$\mathrm{1st}\:\mathrm{case}\::\:{n}\:\mathrm{is}\:\mathrm{odd} \\…
Question Number 116500 by bobhans last updated on 04/Oct/20 $$\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{\mathrm{sin}\:^{\mathrm{3}} \left(\mathrm{x}\right)}{\mathrm{x}}\:\mathrm{dx}\:=? \\ $$ Answered by Olaf last updated on 04/Oct/20 $$\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}^{\mathrm{3}}…
Question Number 116501 by bemath last updated on 04/Oct/20 $$\int\:\frac{\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}}{\:\sqrt{\mathrm{sin}\:\mathrm{2x}}}\:\mathrm{dx}\:? \\ $$ Answered by TANMAY PANACEA last updated on 04/Oct/20 $$\frac{{d}}{{dx}}\left({sinx}+{cosx}\right)={cosx}−{sinx} \\ $$$$\int\frac{−{d}\left({sinx}+{cosx}\right)}{\:\sqrt{\mathrm{1}−\mathrm{1}+{sin}\mathrm{2}{x}}} \\ $$$$\int\frac{−{d}\left({sinx}+{cosx}\right)}{\:\sqrt{\left({sinx}+{cosx}\right)^{\mathrm{2}}…
Question Number 116436 by mnjuly1970 last updated on 04/Oct/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…\:{advanced}\:\:{calculus}… \\ $$$$\:\:\:\:\:\:{evaluate}\::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{I}=\:\int_{\:\mathrm{0}} ^{\:\:\infty} \:\:\left(\frac{{sin}\left({x}\right).{sin}\left(\mathrm{2}{x}\right)}{{x}}\right)\:{dx}\:=???\: \\ $$$$\:\:\:\:\:\:\:\:\:…\:{m}.{n}.\mathrm{1970}… \\ $$$$\: \\ $$$$ \\…