Question Number 11865 by ankhaaankhaa last updated on 03/Apr/17 $$\underset{\mathrm{2}} {\overset{\mathrm{4}} {\int}}\sqrt{\mathrm{16}−{x}^{\mathrm{2}} }{dx}/{x}^{\mathrm{4}} = \\ $$ Answered by ajfour last updated on 03/Apr/17 $${I}=\:\int_{\mathrm{2}} ^{\mathrm{4}}…
Question Number 142917 by ERA last updated on 07/Jun/21 $$\int\left(\boldsymbol{\mathrm{sin}}^{\mathrm{7}} \left(\boldsymbol{\mathrm{x}}\right)\right)\boldsymbol{\mathrm{dx}} \\ $$ Answered by Ar Brandon last updated on 07/Jun/21 $$\mathrm{2isin}\left(\mathrm{nx}\right)=\mathrm{z}^{\mathrm{n}} −\frac{\mathrm{1}}{\mathrm{z}^{\mathrm{n}} },\:\mathrm{2isinx}=\mathrm{z}−\frac{\mathrm{1}}{\mathrm{z}} \\…
Question Number 142910 by BHOOPENDRA last updated on 07/Jun/21 Answered by qaz last updated on 07/Jun/21 $$\mathrm{y}\left(\mathrm{t}\right)=\mathrm{t}+\mathrm{e}^{−\mathrm{2t}} +\int_{\mathrm{0}} ^{\mathrm{t}} \mathrm{y}\left(\tau\right)\mathrm{e}^{\mathrm{2}\left(\mathrm{t}−\tau\right)} \mathrm{d}\tau \\ $$$$\mathscr{L}=\frac{\mathrm{1}}{\mathrm{s}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{s}+\mathrm{2}}+\frac{\mathscr{L}}{\mathrm{s}−\mathrm{2}}…………..\mathscr{L}=\mathscr{L}\left(\mathrm{y}\left(\mathrm{t}\right)\right)\left(\mathrm{s}\right) \\…
Question Number 142900 by aliibrahim1 last updated on 06/Jun/21 Commented by qaz last updated on 07/Jun/21 $$\mathrm{ln}\left(\mathrm{2sin}\:\frac{\mathrm{x}}{\mathrm{2}}\right)=−\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{cos}\:\left(\mathrm{nx}\right)}{\mathrm{n}} \\ $$ Commented by mathmax by…
Question Number 142893 by mnjuly1970 last updated on 06/Jun/21 $$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:…..{mathematical}\:…..{analysis}…… \\ $$$$\:\:\:\:\:\:\:{f}\:\in\:{C}\:\left[\mathrm{0},\mathrm{1}\right]\:{and}\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{{n}} {f}\left({x}\right){dx}=\frac{\mathrm{1}}{{n}+\mathrm{2}}\:,\:{n}\in\mathbb{N} \\ $$$$\:\:\:\:\:\:\:\:{prove}\:\:{f}\left({x}\right):={x}\:….. \\ $$ Answered by mindispower last…
Question Number 11822 by Joel576 last updated on 01/Apr/17 $$\int\:\frac{\mathrm{tan}\:{x}}{\mathrm{1}\:+\:\mathrm{sin}\:{x}}\:{dx} \\ $$ Answered by ajfour last updated on 01/Apr/17 $${let}\:{x}=\frac{\pi}{\mathrm{2}}−\theta \\ $$$${I}=\int\frac{\mathrm{tan}\:{x}}{\mathrm{1}+\mathrm{sin}\:{x}}{dx}\:=\int\frac{−{d}\theta}{\mathrm{tan}\:\theta\:\left(\mathrm{1}+\mathrm{cos}\:\theta\:\right)} \\ $$$$=−\int\frac{\:\mathrm{1}−\mathrm{tan}\:^{\mathrm{2}} \left(\frac{\theta}{\mathrm{2}}\right)}{\mathrm{2tan}\left(\frac{\theta}{\mathrm{2}}\right)\mathrm{2cos}\:^{\mathrm{2}}…
Question Number 142875 by Snail last updated on 06/Jun/21 $${Prove}\:{that}\:\zeta\left({s}\right)=\underset{{prime}} {\prod}\:\frac{\mathrm{1}}{\mathrm{1}−{p}^{−{s}} } \\ $$ Answered by Dwaipayan Shikari last updated on 06/Jun/21 $$\zeta\left({s}\right)=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}^{{s}} }+\frac{\mathrm{1}}{\mathrm{3}^{{s}} }+\frac{\mathrm{1}}{\mathrm{4}^{{s}}…
Question Number 11800 by tawa last updated on 01/Apr/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum} \\ $$$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\left(\mathrm{n}\:+\:\mathrm{1}\right)\sqrt{\mathrm{n}}\:\:+\:\:\mathrm{n}\sqrt{\mathrm{n}\:+\:\mathrm{1}}} \\ $$ Answered by FilupS last updated on 01/Apr/17 $$\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)\sqrt{{n}}+{n}\sqrt{{n}+\mathrm{1}}}=\frac{\mathrm{1}}{\:\sqrt{{n}}}−\frac{\mathrm{1}}{\:\sqrt{{n}+\mathrm{1}}} \\…
Question Number 77335 by aliesam last updated on 05/Jan/20 Commented by aliesam last updated on 05/Jan/20 Commented by msup trace by abdo last updated on…
Question Number 77323 by Boyka last updated on 05/Jan/20 Commented by Tony Lin last updated on 05/Jan/20 $$\int\frac{\mathrm{1}}{\:\sqrt{{cos}\left({x}−\frac{\pi}{\mathrm{2}}\right)}}{dx} \\ $$$$=\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−\mathrm{2}{sin}^{\mathrm{2}} \left(\frac{\mathrm{2}{x}−\pi}{\mathrm{4}}\right)}}{dx} \\ $$$${let}\:{u}=\frac{\mathrm{2}{x}−\pi}{\mathrm{4}},\:\frac{{du}}{{dx}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{2}\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−\mathrm{2}{sin}^{\mathrm{2}}…