Question-132287

Question Number 132287 by benjo_mathlover last updated on 13/Feb/21

Answered by Olaf last updated on 13/Feb/21

Let q = 2∫_0 ^6 f(x)dx = 2∫_0 ^6 f(x−4)dx (1) Let u = x−4 (1) : q = 2∫_(−4) ^2 f(u)du q = 2∫_(−4) ^(−2) f(u)du+2∫_(−2) ^0 f(u)du+2∫_0 ^2 f(u)du f is odd : q = −2∫_2 ^4 f(u)du−2∫_0 ^2 f(u)du+2∫_0 ^2 f(u)du q = −2∫_2 ^4 f(u−4)du Let v = u−4 q = −2∫_(−2) ^0 f(v)dv f is odd : q = 2∫_0 ^2 f(v)dv q = 2p

$$\mathrm{Let}\:{q}\:=\:\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{6}} {f}\left({x}\right){dx}\:=\:\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{6}} {f}\left({x}−\mathrm{4}\right){dx}\:\:\:\:\left(\mathrm{1}\right) \\ $$$$\mathrm{Let}\:{u}\:=\:{x}−\mathrm{4} \\ $$$$\left(\mathrm{1}\right)\::\:{q}\:=\:\mathrm{2}\int_{−\mathrm{4}} ^{\mathrm{2}} {f}\left({u}\right){du} \\ $$$${q}\:=\:\mathrm{2}\int_{−\mathrm{4}} ^{−\mathrm{2}} {f}\left({u}\right){du}+\mathrm{2}\int_{−\mathrm{2}} ^{\mathrm{0}} {f}\left({u}\right){du}+\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{2}} {f}\left({u}\right){du} \\ $$$${f}\:\mathrm{is}\:\mathrm{odd}\:: \\ $$$${q}\:=\:−\mathrm{2}\int_{\mathrm{2}} ^{\mathrm{4}} {f}\left({u}\right){du}−\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{2}} {f}\left({u}\right){du}+\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{2}} {f}\left({u}\right){du} \\ $$$${q}\:=\:−\mathrm{2}\int_{\mathrm{2}} ^{\mathrm{4}} {f}\left({u}−\mathrm{4}\right){du} \\ $$$$\mathrm{Let}\:{v}\:=\:{u}−\mathrm{4} \\ $$$${q}\:=\:−\mathrm{2}\int_{−\mathrm{2}} ^{\mathrm{0}} {f}\left({v}\right){dv} \\ $$$${f}\:\mathrm{is}\:\mathrm{odd}\:: \\ $$$${q}\:=\:\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{2}} {f}\left({v}\right){dv} \\ $$$${q}\:=\:\mathrm{2}{p} \\ $$

Commented by Olaf last updated on 13/Feb/21

$${sorry}\:{mister},\:{I}\:{corrected} \\ $$

Commented by benjo_mathlover last updated on 13/Feb/21

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir} \\ $$

Answered by Ñï= last updated on 14/Feb/21

2∫_0 ^6 f(x)dx =2∫_0 ^6 f(x−4)dx =2∫_(−4) ^2 f(x)dx =2(∫_(−2) ^2 f(x)dx+∫_(−4) ^(−2) f(x)dx) =2∫_0 ^2 f(x−4)dx =2∫_0 ^2 f(x)dx =2p

$$\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{6}} {f}\left({x}\right){dx} \\ $$$$=\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{6}} {f}\left({x}−\mathrm{4}\right){dx} \\ $$$$=\mathrm{2}\int_{−\mathrm{4}} ^{\mathrm{2}} {f}\left({x}\right){dx} \\ $$$$=\mathrm{2}\left(\int_{−\mathrm{2}} ^{\mathrm{2}} {f}\left({x}\right){dx}+\int_{−\mathrm{4}} ^{−\mathrm{2}} {f}\left({x}\right){dx}\right) \\ $$$$=\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{2}} {f}\left({x}−\mathrm{4}\right){dx} \\ $$$$=\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{2}} {f}\left({x}\right){dx} \\ $$$$=\mathrm{2}{p} \\ $$

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