Question Number 39441 by rahul 19 last updated on 06/Jul/18
$$\int_{\frac{\mathrm{1}}{\mathrm{4}}} ^{\:\mathrm{4}} \:\frac{\mathrm{1}}{{x}}\:\mathrm{sin}\:\left({x}−\frac{\mathrm{1}}{{x}}\right){dx}\:=\:? \\ $$
Commented by prof Abdo imad last updated on 06/Jul/18
$${changement}\:\:{x}=\frac{\mathrm{1}}{{t}}\:\:{give} \\ $$$${I}\:\:=\:−\int_{\frac{\mathrm{1}}{\mathrm{4}}} ^{\mathrm{4}} \:{tsin}\left(\frac{\mathrm{1}}{{t}}\:−{t}\right)\frac{−{dt}}{{t}^{\mathrm{2}} } \\ $$$$=\:\int_{\frac{\mathrm{1}}{\mathrm{4}}} ^{\mathrm{4}} \:\frac{\mathrm{1}}{{t}}{sin}\left(\frac{\mathrm{1}}{{t}}−{t}\right){dt}\:=\:−\int_{\frac{\mathrm{1}}{\mathrm{4}}} ^{\mathrm{4}} \:\:\frac{\mathrm{1}}{{t}}{sin}\left({t}−\frac{\mathrm{1}}{{t}}\right){dt} \\ $$$$=−{I}\:\Rightarrow\:\mathrm{2}{I}\:=\mathrm{0}\:\Rightarrow\:{I}\:=\mathrm{0} \\ $$
Commented by rahul 19 last updated on 06/Jul/18
Thanks prof Abdo ����