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Question Number 17220 by Arnab Maiti last updated on 02/Jul/17
Show that ∫_a ^( b) f(kx)dx=(1/k)∫_(ka) ^( kb) f(x)dx
$$\mathrm{Show}\:\mathrm{that}\:\int_{\mathrm{a}} ^{\:\mathrm{b}} {f}\left(\mathrm{kx}\right)\mathrm{dx}=\frac{\mathrm{1}}{\mathrm{k}}\int_{\mathrm{ka}} ^{\:\mathrm{kb}} {f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$
Answered by ajfour last updated on 02/Jul/17
let kx=t   ⇒   dx=(dt/k)  and when x=a  ,  t=ka        when x=b,  t=kb  so  ∫_a ^(  b) f(kx)dx= (1/k)∫_( ka) ^(  kb) f(t)dt                      =(1/k)∫_( ka) ^(  kb) f(x)dx .
$$\mathrm{let}\:\mathrm{kx}=\mathrm{t}\:\:\:\Rightarrow\:\:\:\mathrm{dx}=\frac{\mathrm{dt}}{\mathrm{k}} \\ $$$$\mathrm{and}\:\mathrm{when}\:\mathrm{x}=\mathrm{a}\:\:,\:\:\mathrm{t}=\mathrm{ka} \\ $$$$\:\:\:\:\:\:\mathrm{when}\:\mathrm{x}=\mathrm{b},\:\:\mathrm{t}=\mathrm{kb} \\ $$$$\mathrm{so}\:\:\int_{\mathrm{a}} ^{\:\:\mathrm{b}} \mathrm{f}\left(\mathrm{kx}\right)\mathrm{dx}=\:\frac{\mathrm{1}}{\mathrm{k}}\int_{\:\mathrm{ka}} ^{\:\:\mathrm{kb}} \mathrm{f}\left(\mathrm{t}\right)\mathrm{dt} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{k}}\int_{\:\mathrm{ka}} ^{\:\:\mathrm{kb}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\:. \\ $$
Commented by Arnab Maiti last updated on 02/Jul/17
Thank u sir.
$$\mathrm{Thank}\:\mathrm{u}\:\mathrm{sir}. \\ $$

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