Question Number 139248 by ajfour last updated on 24/Apr/21
$$\int_{{a}} ^{\:{b}} {f}\left({t}\right){g}'\left({t}\right){dt}=\:? \\ $$
Answered by mathmax by abdo last updated on 25/Apr/21
$$\int_{\mathrm{a}} ^{\mathrm{b}} \:\mathrm{f}\left(\mathrm{t}\right)\mathrm{g}^{'} \left(\mathrm{t}\right)\mathrm{dt}\:=\left[\mathrm{f}\left(\mathrm{t}\right)\mathrm{g}\left(\mathrm{t}\right)\right]_{\mathrm{a}} ^{\mathrm{b}} −\int_{\mathrm{a}} ^{\mathrm{b}} \:\mathrm{f}^{'} \left(\mathrm{t}\right)\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt} \\ $$$$=\mathrm{f}\left(\mathrm{b}\right)\mathrm{g}\left(\mathrm{b}\right)−\mathrm{f}\left(\mathrm{a}\right)\mathrm{g}\left(\mathrm{a}\right)−\int_{\mathrm{a}} ^{\mathrm{b}} \:\mathrm{f}'\left(\mathrm{t}\right)\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt} \\ $$
Answered by physicstutes last updated on 24/Apr/21
$${u}\:=\:{f}\left({t}\right)\:\mathrm{and}\:{dv}\:=\:\mathrm{g}'\left({t}\right){dt} \\ $$$$\Rightarrow\:{du}\:=\:{f}\:'\left({t}\right){dt}\:\mathrm{and}\:{v}\:=\:\mathrm{g}\left({t}\right) \\ $$$$\Rightarrow\:\left[\mathrm{g}\left({t}\right)\:{f}\left({t}\right)\right]_{{a}} ^{{b}} −\int_{{a}} ^{{b}} \mathrm{g}\left({t}\right)\:{f}'\left({t}\right){dt} \\ $$$${I}\:=\:\int_{{a}} ^{{b}} \mathrm{g}\left({t}\right){f}\:'\left({t}\right)\:{dt}\:=\:\left[\mathrm{g}\left({t}\right){f}\left({t}\right)\right]_{{a}} ^{{b}} −\int_{{a}} ^{{b}} {f}\left({t}\right)\mathrm{g}'\left({t}\right){dt} \\ $$$$\Rightarrow\:\int_{{a}} ^{{b}} {f}\left({t}\right)\mathrm{g}'\left({t}\right){dt}\:=\:\left[\mathrm{g}\left({t}\right){f}\left({t}\right)\right]_{{a}} ^{{b}} −\left\{\left[\mathrm{g}\left({t}\right){f}\left({t}\right)\right]_{{a}} ^{{b}} −\int_{{a}} ^{{b}} {f}\left({t}\right)\mathrm{g}'\left({t}\right){dt}\right\} \\ $$$$\:\begin{array}{|c|}{\int_{{a}} ^{{b}} {f}\left({t}\right)\mathrm{g}'\left({t}\right){dt}\:=\:\mathrm{0}}\\\hline\end{array}\begin{array}{|c|c|}\\\\\hline\end{array} \\ $$
Commented by mr W last updated on 24/Apr/21
$${wrong}\:{sir}! \\ $$$${you}\:{showed}\:{only} \\ $$$$\int_{{a}} ^{{b}} {f}\left({t}\right)\mathrm{g}'\left({t}\right){dt}=\int_{{a}} ^{{b}} {f}\left({t}\right)\mathrm{g}'\left({t}\right){dt} \\ $$
Commented by mr W last updated on 24/Apr/21
$${if}\:\int_{{a}} ^{{b}} {f}\left({t}\right)\mathrm{g}'\left({t}\right){dt}=\mathrm{0},\:{then}\:{it}\:{means} \\ $$$$\int_{{a}} ^{{b}} {f}\left({t}\right){dt}=\mathrm{0},\:{because}\:{you}\:{can}\:{choose} \\ $$$${g}\left({t}\right)={t}\:{and}\:{g}'\left({t}\right)=\mathrm{1}. \\ $$