Question Number 51 by surabhi last updated on 25/Jan/15 $$\mathrm{Evaluate}\:\int_{\mathrm{0}} ^{\pi/\mathrm{2}} {x}\mathrm{cos}\:{x}\:{dx} \\ $$ Answered by surabhi last updated on 04/Nov/14 $$\int_{\mathrm{0}} ^{\pi/\mathrm{2}} {x}\mathrm{cos}\:{x}\:{dx}=\left[{x}\mathrm{sin}\:{x}\right]_{\mathrm{0}} ^{\pi/\mathrm{2}}…
Question Number 46 by surabhi last updated on 25/Jan/15 $$\int\left(\mathrm{log}\:{x}\right)^{\mathrm{2}} {dx} \\ $$ Answered by surabhi last updated on 04/Nov/14 $$\int\left(\mathrm{log}\:{x}\right)^{\mathrm{2}} {dx}=\int\mid\left(\mathrm{log}\:{x}\right)^{\mathrm{2}} \centerdot\mathrm{1}\mid{dx} \\ $$$$=\left(\mathrm{log}\:{x}\right)^{\mathrm{2}}…
Question Number 48 by surabhi last updated on 25/Jan/15 $$\mathrm{Evaluate}\:\:\:\int\sqrt{\mathrm{3}−\mathrm{2}{x}−\mathrm{2}{x}^{\mathrm{2}} \:}{dx}. \\ $$ Answered by surabhi last updated on 04/Nov/14 $$\int\sqrt{\mathrm{3}−\mathrm{2}{x}−\mathrm{2}{x}^{\mathrm{2}} }\:{dx} \\ $$$$=\sqrt{\mathrm{2}}\:\centerdot\:\int\sqrt{\left(\frac{\mathrm{3}}{\mathrm{2}}−{x}−{x}^{\mathrm{2}} \right)}\:{dx}…
Question Number 50 by surabhi last updated on 25/Jan/15 $$\mathrm{Evaluate}\:\:\:\int\left({x}+\mathrm{1}\right)\sqrt{\mathrm{1}−{x}−{x}^{\mathrm{2}} }{dx}. \\ $$ Commented by 123456 last updated on 13/Dec/14 $$\mathrm{tente}?\mathrm{completar}\:\mathrm{quadrados}\:\mathrm{e}\:\mathrm{fca}\:\mathrm{uma}\:\mathrm{substutuico} \\ $$ Answered by…
Question Number 40 by user3 last updated on 25/Jan/15 $$\mathrm{Evaluate}\:\:\:\int{x}\:\mathrm{cos}\:^{\mathrm{2}} {x}\:{dx}. \\ $$ Answered by user3 last updated on 03/Nov/14 $$\int{x}\:\mathrm{cos}^{\mathrm{2}} {x}\:{dx}\:=\int{x}\left(\frac{\mathrm{1}+\mathrm{cos}\:\mathrm{2}{x}}{\mathrm{2}}\right){dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int{x}\:{dx}+\frac{\mathrm{1}}{\mathrm{2}}\int{x}\:\mathrm{cos}\:\mathrm{2}{x}\:{dx} \\…
Question Number 65581 by aliesam last updated on 31/Jul/19 Commented by mathmax by abdo last updated on 31/Jul/19 $${let}\:{A}\:=\int\:{e}^{\left(\frac{\mathrm{1}}{{x}}−{x}\right)} \:{dx}\:\:\:{we}\:{have}\:{e}^{{u}} \:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{u}^{{n}} }{{n}!}\:\:{with}\:{radius}\:{infinite}\Rightarrow \\…
Question Number 35 by user2 last updated on 25/Jan/15 $$\mathrm{Evaluate}\:\:\int{x}^{\mathrm{2}} \mathrm{sin}\:{x}\:{dx} \\ $$ Answered by surabhi last updated on 04/Nov/14 $$\int{x}^{\mathrm{2}} \mathrm{sin}\:{xdx}= \\ $$$$={x}^{\mathrm{2}} \int\mathrm{sin}\:{x}\:{dx}−\int\left[\frac{{d}}{{dx}}\left({x}^{\mathrm{2}}…
Question Number 33 by user2 last updated on 25/Jan/15 $$\mathrm{Evaluate}\:\:\int\mathrm{sin}\:{x}\:\sqrt{\mathrm{1}−\mathrm{cos}\:\mathrm{2}{x}\:}{dx}. \\ $$ Answered by user2 last updated on 03/Nov/14 $$\int\mathrm{sin}\:{x}\:\sqrt{\mathrm{1}−\mathrm{cos}\:\mathrm{2}{x}}{dx} \\ $$$$=\int\mathrm{sin}\:{x}\centerdot\sqrt{\mathrm{2sin}^{\mathrm{2}} {x}\:}{dx}\: \\ $$$$=\sqrt{\mathrm{2}}\int\mathrm{sin}^{\mathrm{2}}…
Question Number 31 by user2 last updated on 25/Jan/15 $$\mathrm{Evaluate}\:\int\sqrt{\mathrm{1}+\mathrm{sin}\:{x}}{dx}. \\ $$ Answered by user2 last updated on 03/Nov/14 $$\int\sqrt{\mathrm{1}+\mathrm{sin}\:{x}}\:{dx}=\int\sqrt{\mathrm{sin}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}+\mathrm{cos}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}+\mathrm{2sin}\frac{{x}}{\mathrm{2}}\mathrm{cos}\frac{{x}}{\mathrm{2}}\:}{dx} \\ $$$$=\int\left(\mathrm{sin}\frac{{x}}{\mathrm{2}}+\mathrm{cos}\frac{{x}}{\mathrm{2}}\right){dx} \\…
Question Number 131073 by Algoritm last updated on 01/Feb/21 Answered by mr W last updated on 01/Feb/21 $${F}\:'\left({x}\right)={e}^{\left(\mathrm{ln}\:{x}\right)^{\mathrm{100}} } \frac{\mathrm{1}}{{x}} \\ $$$${F}\:'\left({e}\right)={e}^{\left(\mathrm{ln}\:{e}\right)^{\mathrm{100}} } \frac{\mathrm{1}}{{e}}={e}×\frac{\mathrm{1}}{{e}}=\mathrm{1} \\…