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0-2-ln-3-xe-x-2-2-dx-




Question Number 94948 by rb222 last updated on 22/May/20
∫_0 ^((√(2 ln (3))) ) xe^(x^2 /2)  dx = . . .
$$\int_{\mathrm{0}} ^{\sqrt{\mathrm{2}\:{ln}\:\left(\mathrm{3}\right)}\:} {xe}^{\frac{{x}^{\mathrm{2}} }{\mathrm{2}}} \:{dx}\:=\:.\:.\:. \\ $$
Commented by Tony Lin last updated on 22/May/20
let (x^2 /2)=u, du=xdx  ∫_0 ^(ln3) e^u du  =3−1=2
$${let}\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}}={u},\:{du}={xdx} \\ $$$$\int_{\mathrm{0}} ^{{ln}\mathrm{3}} {e}^{{u}} {du} \\ $$$$=\mathrm{3}−\mathrm{1}=\mathrm{2} \\ $$
Commented by rb222 last updated on 22/May/20
thanks sir
$${thanks}\:{sir} \\ $$
Commented by john santu last updated on 22/May/20
∫ x.e^(x^2 /2)  dx = ∫e^((1/2)x^2 )  d((1/2)x^2 )  = e^((1/2)x^2 )   then ∫_0 ^(√(2 ln(3))) x.e^((1/2)x^2 )  dx = [ e^((1/2)x^2 )  ]_( 0) ^(√(2 ln(3)))   = e^(ln(3)) −e^0  = 3−1 = 2
$$\int\:{x}.{e}^{\frac{{x}^{\mathrm{2}} }{\mathrm{2}}} \:{dx}\:=\:\int{e}^{\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} } \:{d}\left(\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} \right) \\ $$$$=\:{e}^{\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} } \\ $$$${then}\:\underset{\mathrm{0}} {\overset{\sqrt{\mathrm{2}\:\mathrm{ln}\left(\mathrm{3}\right)}} {\int}}{x}.{e}^{\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} } \:{dx}\:=\:\left[\:{e}^{\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} } \:\right]_{\:\mathrm{0}} ^{\sqrt{\mathrm{2}\:\mathrm{ln}\left(\mathrm{3}\right)}} \\ $$$$=\:{e}^{\mathrm{ln}\left(\mathrm{3}\right)} −\mathrm{e}^{\mathrm{0}} \:=\:\mathrm{3}−\mathrm{1}\:=\:\mathrm{2} \\ $$
Answered by niroj last updated on 22/May/20
  ∫_0 ^( (√(2In(3)))) x e^(x^2 /2) dx    Put, (x^2 /2)= t          x^2 = 2t        2xdx=2dt          xdx=dt        If x=(√(2ln(3)))  ⇒t =((2In(3))/2)=In(3)        If x=0 ⇒ t=0     ∫_0 ^( In(3))  e^t dt   = [ e^t ]_0 ^(ln(3))    =e^(ln(3)) −e^0     =3−1= 2 //.
$$\:\:\int_{\mathrm{0}} ^{\:\sqrt{\mathrm{2}{In}\left(\mathrm{3}\right)}} \mathrm{x}\:\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}} \mathrm{dx} \\ $$$$\:\:\mathrm{Put},\:\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}=\:\mathrm{t} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{x}^{\mathrm{2}} =\:\mathrm{2t} \\ $$$$\:\:\:\:\:\:\mathrm{2xdx}=\mathrm{2dt} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{xdx}=\mathrm{dt} \\ $$$$\:\:\:\:\:\:\mathrm{If}\:\mathrm{x}=\sqrt{\mathrm{2ln}\left(\mathrm{3}\right)}\:\:\Rightarrow\mathrm{t}\:=\frac{\mathrm{2In}\left(\mathrm{3}\right)}{\mathrm{2}}=\mathrm{In}\left(\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\mathrm{If}\:\mathrm{x}=\mathrm{0}\:\Rightarrow\:\mathrm{t}=\mathrm{0} \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{In}\left(\mathrm{3}\right)} \:\mathrm{e}^{\mathrm{t}} \mathrm{dt} \\ $$$$\:=\:\left[\:\mathrm{e}^{\mathrm{t}} \right]_{\mathrm{0}} ^{\mathrm{ln}\left(\mathrm{3}\right)} \\ $$$$\:=\mathrm{e}^{\mathrm{ln}\left(\mathrm{3}\right)} −\mathrm{e}^{\mathrm{0}} \\ $$$$\:\:=\mathrm{3}−\mathrm{1}=\:\mathrm{2}\://. \\ $$$$ \\ $$
Commented by rb222 last updated on 22/May/20
thanks sir
$${thanks}\:{sir} \\ $$
Commented by niroj last updated on 22/May/20
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Commented by peter frank last updated on 22/May/20
thank you
$$\mathrm{thank}\:\mathrm{you} \\ $$

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