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Category: Integration

a-particle-starts-with-an-initial-speed-u-it-moves-in-a-straight-line-with-an-accleration-which-varies-as-the-square-of-the-time-the-particle-has-been-in-motion-Find-the-speed-at-any-time-t-and-the-d

Question Number 17599 by chux last updated on 08/Jul/17 $$\mathrm{a}\:\mathrm{particle}\:\mathrm{starts}\:\mathrm{with}\:\mathrm{an}\:\mathrm{initial} \\ $$$$\mathrm{speed}\:\mathrm{u},\mathrm{it}\:\mathrm{moves}\:\mathrm{in}\:\mathrm{a}\:\mathrm{straight} \\ $$$$\mathrm{line}\:\mathrm{with}\:\mathrm{an}\:\mathrm{accleration}\:\mathrm{which} \\ $$$$\mathrm{varies}\:\mathrm{as}\:\mathrm{the}\:\mathrm{square}\:\mathrm{of}\:\mathrm{the}\:\mathrm{time} \\ $$$$\mathrm{the}\:\mathrm{particle}\:\mathrm{has}\:\mathrm{been}\:\mathrm{in}\:\mathrm{motion}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{at}\:\mathrm{any}\:\mathrm{time}\:\mathrm{t},\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{distance}\:\mathrm{travelled}. \\ $$ Answered…

x-2-1-x-2-1-x-2-2x-2-dx-

Question Number 83123 by M±th+et£s last updated on 28/Feb/20 $$\int\frac{\left({x}^{\mathrm{2}} −\mathrm{1}\right)}{\left(\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\right)\left({x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{2}\right)}\:{dx} \\ $$ Commented by mathmax by abdo last updated on 28/Feb/20 $${A}\:=\int\:\:\frac{{x}^{\mathrm{2}}…

cos-xe-sin-x-dx-

Question Number 83115 by 09658867628 last updated on 28/Feb/20 $$\int\mathrm{cos}\:{xe}^{\mathrm{sin}\:{x}} {dx} \\ $$ Commented by niroj last updated on 28/Feb/20 $$ \\ $$$$\:\int\boldsymbol{\mathrm{cos}}\:\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{sin}}\:\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{dx}} \\…

prove-that-0-pi-4-cos-nx-cos-n-x-dx-2-n-pi-8-k-1-n-1-sin-kpi-4-2k-2-k-n-N-

Question Number 83108 by M±th+et£s last updated on 28/Feb/20 $${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{{cos}\left({nx}\right)}{{cos}^{{n}} \left({x}\right)}\:{dx}\:=\mathrm{2}^{{n}} \left[\frac{\pi}{\mathrm{8}}−\underset{{k}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\sum}}\frac{{sin}\left(\frac{{k}\pi}{\mathrm{4}}\right)}{\mathrm{2}{k}\left(\sqrt{\mathrm{2}}\right)^{{k}} }\right]\:{n}\in{N}^{\ast} \\ $$ Answered by mind is…

bounded-by-the-curve-y-4-x-y-0-y-1-

Question Number 83110 by 09658867628 last updated on 28/Feb/20 $${bounded}\:{by}\:{the}\:{curve}\:{y}=\sqrt{\mathrm{4}-{x}}\:{y}=\mathrm{0}\:{y}=\mathrm{1} \\ $$ Commented by jagoll last updated on 28/Feb/20 $$\mathrm{Area}\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{4}−\mathrm{y}^{\mathrm{2}} \right)\:\mathrm{dy}\: \\ $$$$=\:\mathrm{4y}\:−\:\frac{\mathrm{y}^{\mathrm{3}}…

1-e-e-dt-t-

Question Number 83109 by 09658867628 last updated on 28/Feb/20 $$\int_{\mathrm{1}/\boldsymbol{{e}}} ^{{e}} \frac{\boldsymbol{{dt}}}{\boldsymbol{{t}}} \\ $$ Commented by jagoll last updated on 28/Feb/20 $$=\:\mathrm{ln}\:\left(\mathrm{t}\right)\:\mid_{\frac{\mathrm{1}}{\mathrm{e}}} ^{\mathrm{e}} \:=\:\mathrm{1}−\left(−\mathrm{1}\right)\:=\:\mathrm{2} \\…