Question Number 22166 by Joel577 last updated on 12/Oct/17
$$\mathrm{If}\underset{\mathrm{1}} {\overset{\mathrm{4}} {\int}}\:{f}\left({x}\right)\:{dx}\:=\:\mathrm{5} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:{f}\left(\mathrm{3}{x}\:+\mathrm{1}\right)\:{dx}\:? \\ $$
Answered by ajfour last updated on 12/Oct/17
$${I}=\int_{\mathrm{0}} ^{\:\:\mathrm{1}} {f}\left(\mathrm{3}{x}+\mathrm{1}\right){dx}\: \\ $$$${let}\:\mathrm{3}{x}+\mathrm{1}={t}\:\:\:\Rightarrow\:\:{dx}=\frac{{dt}}{\mathrm{3}} \\ $$$$\:{clearly}\:\:{then}\:{x}=\mathrm{0}\:\:\Rightarrow\:{t}=\mathrm{1} \\ $$$${and}\:{when}\:{x}=\mathrm{1}\:,\:{t}=\mathrm{4} \\ $$$${so}\:\:\:{I}=\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{1}} ^{\:\:\mathrm{4}} {f}\left({t}\right){dt}\:=\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{1}} ^{\:\:\mathrm{4}} {f}\left(\rho\right){d}\rho \\ $$$$\:\:\:\:=\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{1}} ^{\:\:\mathrm{4}} {f}\left({x}\right){dx}\:=\:\frac{\mathrm{1}}{\mathrm{3}}×\mathrm{5}\:=\frac{\mathrm{5}}{\mathrm{3}}\:. \\ $$
Commented by Joel577 last updated on 13/Oct/17
$${thank}\:{you}\:{very}\:{much} \\ $$