Question Number 208205 by efronzo1 last updated on 07/Jun/24 $$\:\:\:\int\:\left({x}^{\mathrm{3}} .\:\mathrm{5}^{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{2}} \:\right)\:{dx}\:=? \\ $$ Answered by Frix last updated on 07/Jun/24 $$\int{x}^{\mathrm{3}} \mathrm{5}^{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{2}}…
Question Number 208194 by universe last updated on 07/Jun/24 $$\:\:\:\:\:\:\:\:{I}_{{n}} \:=\:\:\int_{\mathrm{0}\:} ^{\infty} \frac{\mathrm{1}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }{dx} \\ $$$$\:\:{prove}\:{that}\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{I}_{{n}} }{{n}}\:\:=\:\:\pi \\ $$ Answered by Berbere…
Question Number 208140 by Ghisom last updated on 06/Jun/24 $$\underset{\mathrm{0}} {\overset{\pi} {\int}}\frac{{dx}}{\mathrm{1}+\sqrt[{\mathrm{3}}]{\mathrm{sin}\:{x}}}=? \\ $$$$\mathrm{exact}\:\mathrm{result}\:\mathrm{required} \\ $$ Answered by Frix last updated on 07/Jun/24 $$=\mathrm{2}\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}}…
Question Number 208129 by efronzo1 last updated on 06/Jun/24 $$\:\:\:\:\:\underbrace{\pm \cancel{} } \\ $$ Answered by mr W last updated on 06/Jun/24 $${f}\left(\mathrm{0}\right)=\mathrm{1} \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{379}…
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Question Number 208062 by mathzup last updated on 03/Jun/24 $${find}\:\:\int_{\mathrm{4}} ^{\infty} \:\:\:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{3}} \left({x}−\mathrm{3}\right)^{\mathrm{5}} } \\ $$ Answered by Frix last updated on 03/Jun/24 $$\mathrm{Use}\:\mathrm{Ostrogradski}'\mathrm{s}\:\mathrm{Method}\:\left(\mathrm{search}\:\mathrm{it}\:\mathrm{on}\right. \\…
Question Number 208052 by necx122 last updated on 03/Jun/24 $${Sketch}\:{the}\:{curve}\:{y}\:=\:{x}^{\mathrm{3}} . \\ $$$$\left({a}\right)\:{Find}\:{the}\:{equation}\:{of}\:{the}\:{tangent} \\ $$$${to}\:{the}\:{curve}\:{at}\:{A}\left(\mathrm{1},\mathrm{1}\right). \\ $$$$\left({b}\right)\:{Find}\:{the}\:{coordinates}\:{of}\:{point}\:{B}, \\ $$$${where}\:{the}\:{tangent}\:{meets}\:{the}\:{curve}\:{again}. \\ $$$$\left({c}\right)\:{Calculate}\:{the}\:{area}\:{between}\:{the} \\ $$$${tangent}\:{B}\:{and}\:{the}\:{arc}\:{AB}\:{of}\:{the}\:{curve}. \\ $$…
Question Number 207949 by mnjuly1970 last updated on 31/May/24 $$ \\ $$$$\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:{Find}\:{the}\:{value}\:{of}\:: \\ $$$$ \\ $$$$\:\:\:\:\boldsymbol{\Omega}\:=\:\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} \:\frac{\:\boldsymbol{{dx}}}{\boldsymbol{{sin}}^{\mathrm{6}} \boldsymbol{{x}}\:+\:\boldsymbol{{cos}}^{\mathrm{6}} \boldsymbol{{x}}}\:=\:?\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−−−−−−−−− \\…
Question Number 207938 by necx122 last updated on 31/May/24 $${what}\:{is}\:{the}\:{area}\:{bounded}\:{by}\:{the}\:{curve} \\ $$$${y}={x}\left({x}−\mathrm{2}\right)\left({x}−\mathrm{5}\right)\:{and}\:{the}\:{x}\:{axis}? \\ $$$$ \\ $$ Answered by mr W last updated on 31/May/24 $${A}=\int_{\mathrm{0}}…
Question Number 207924 by AliJumaa last updated on 31/May/24 $$\int{f}\left({x}\right){g}\left({x}\right){dx}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\left(−\mathrm{1}\right)^{{n}} \:\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}}{{h}^{{n}} }\:\underset{{i}={o}} {\overset{{n}} {\sum}}\left[\:\left(−\mathrm{1}\right)^{{i}} \left(\frac{{n}!}{{i}!\left({n}−{i}\right)!}\right){f}\left({x}+\left({n}−{i}\right){h}\right)\right]\:\frac{\mathrm{1}}{{n}!}\underset{{a}} {\overset{{x}} {\int}}\left({x}−{t}\right)^{{n}} {g}\left({t}\right){dt}\: \\ $$$${prove}\:{that}\:{right} \\ $$$${its}\:{a}\:{relation}\:{that}\:{i}\:{have}\:{derrived}…