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Question-107591




Question Number 107591 by hgrocks last updated on 11/Aug/20
Commented by hgrocks last updated on 11/Aug/20
Pls Tell Me Which one is correct  and why
$$\mathrm{Pls}\:\mathrm{Tell}\:\mathrm{Me}\:\mathrm{Which}\:\mathrm{one}\:\mathrm{is}\:\mathrm{correct} \\ $$$$\mathrm{and}\:\mathrm{why}\: \\ $$
Commented by hgrocks last updated on 11/Aug/20
Anyone?
$$\mathrm{Anyone}? \\ $$
Answered by Aziztisffola last updated on 11/Aug/20
put x=αt ⇒dx=αdt⇔dt=(dx/α)  t=0 ⇒x=0 and t→∞⇒x→∞  ∫_0 ^( ∞) f(αt)dt=lim_(x→∞) ∫_0 ^( x) f(x)(dx/α)  =(1/α)lim_(x→∞) ∫_0 ^( x) f(x)dx  then (1) is correct
$$\mathrm{put}\:\mathrm{x}=\alpha\mathrm{t}\:\Rightarrow\mathrm{dx}=\alpha\mathrm{dt}\Leftrightarrow\mathrm{dt}=\frac{\mathrm{dx}}{\alpha} \\ $$$$\mathrm{t}=\mathrm{0}\:\Rightarrow\mathrm{x}=\mathrm{0}\:\mathrm{and}\:\mathrm{t}\rightarrow\infty\Rightarrow\mathrm{x}\rightarrow\infty \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \mathrm{f}\left(\alpha\mathrm{t}\right)\mathrm{dt}=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{0}} ^{\:\mathrm{x}} \mathrm{f}\left(\mathrm{x}\right)\frac{\mathrm{dx}}{\alpha} \\ $$$$=\frac{\mathrm{1}}{\alpha}\underset{\mathrm{x}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{0}} ^{\:\mathrm{x}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$$$\mathrm{then}\:\left(\mathrm{1}\right)\:\mathrm{is}\:\mathrm{correct}\: \\ $$
Answered by mathmax by abdo last updated on 11/Aug/20
∫_0 ^∞ f(αt)dt =lim_(n→+∞) ∫_0 ^n f(αt)dt =_(αt =x)   lim_(n→+∞) ∫_0 ^(nα) f(x)(dx/α)  =(1/α) lim_(n→+∞)  ∫_0 ^(nα) f(x)dx  .
$$\int_{\mathrm{0}} ^{\infty} \mathrm{f}\left(\alpha\mathrm{t}\right)\mathrm{dt}\:=\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \int_{\mathrm{0}} ^{\mathrm{n}} \mathrm{f}\left(\alpha\mathrm{t}\right)\mathrm{dt}\:=_{\alpha\mathrm{t}\:=\mathrm{x}} \:\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \int_{\mathrm{0}} ^{\mathrm{n}\alpha} \mathrm{f}\left(\mathrm{x}\right)\frac{\mathrm{dx}}{\alpha} \\ $$$$=\frac{\mathrm{1}}{\alpha}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\mathrm{n}\alpha} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\:\:. \\ $$

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