Xét hàm số f(x) có đạo hàm liên tục trên R và thỏa mãn điều kiện f(1) = 1 ; f(2) = 44. Tính \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 % da9maapehabaWaaeWaaeaadaWcaaqaaiqadAgagaqbamaabmaabaGa % amiEaaGaayjkaiaawMcaaiabgUcaRiaaikdaaeaacaWG4baaaiabgk % HiTmaalaaabaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiab % gUcaRiaaigdaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaaaaaOGaay % jkaiaawMcaaiaabsgacaWG4baaleaacaaIXaaabaGaaGOmaaqdcqGH % RiI8aaaa!4D39! J = \int\limits_1^2 {\left( {\frac{{f'\left( x \right) + 2}}{x} - \frac{{f\left( x \right) + 1}}{{{x^2}}}} \right){\rm{d}}x} \)

Ta có \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 % da9maapehabaWaaeWaaeaadaWcaaqaaiqadAgagaqbamaabmaabaGa % amiEaaGaayjkaiaawMcaaiabgUcaRiaaikdaaeaacaWG4baaaiabgk % HiTmaalaaabaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiab % gUcaRiaaigdaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaaaaaOGaay % jkaiaawMcaaiaabsgacaWG4baaleaacaaIXaaabaGaaGOmaaqdcqGH % RiI8aaaa!4D39! J = \int\limits_1^2 {\left( {\frac{{f'\left( x \right) + 2}}{x} - \frac{{f\left( x \right) + 1}}{{{x^2}}}} \right){\rm{d}}x} \) \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaa8 % qCaeaadaWcaaqaaiqadAgagaqbamaabmaabaGaamiEaaGaayjkaiaa % wMcaaaqaaiaadIhaaaGaaeizaiaadIhacqGHsisldaWdXbqaamaala % aabaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaaqaaiaadIha % daahaaWcbeqaaiaaikdaaaaaaOGaaeizaiaadIhaaSqaaiaaigdaae % aacaaIYaaaniabgUIiYdaaleaacaaIXaaabaGaaGOmaaqdcqGHRiI8 % aOGaey4kaSYaa8qCaeaadaqadaqaamaalaaabaGaaGOmaaqaaiaadI % haaaGaeyOeI0YaaSaaaeaacaaIXaaabaGaamiEamaaCaaaleqabaGa % aGOmaaaaaaaakiaawIcacaGLPaaacaqGKbGaamiEaaWcbaGaaGymaa % qaaiaaikdaa0Gaey4kIipaaaa!5B14! = \int\limits_1^2 {\frac{{f'\left( x \right)}}{x}{\rm{d}}x - \int\limits_1^2 {\frac{{f\left( x \right)}}{{{x^2}}}{\rm{d}}x} } + \int\limits_1^2 {\left( {\frac{2}{x} - \frac{1}{{{x^2}}}} \right){\rm{d}}x} \)

Đặt \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabe % qaaiaadwhacqGH9aqpdaWcaaqaaiaaigdaaeaacaWG4baaaaqaaiaa % bsgacaWG2bGaeyypa0JabmOzayaafaWaaeWaaeaacaWG4baacaGLOa % GaayzkaaGaaeizaiaadIhaaaGaay5EaaGaeyO0H49aaiqaaqaabeqa % aiaabsgacaWG1bGaeyypa0JaeyOeI0YaaSaaaeaacaaIXaaabaGaam % iEamaaCaaaleqabaGaaGOmaaaaaaGccaqGKbGaamiEaaqaaiaadAha % cqGH9aqpcaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaaaaiaawU % haaaaa!5489! \left\{ \begin{array}{l} u = \frac{1}{x}\\ {\rm{d}}v = f'\left( x \right){\rm{d}}x \end{array} \right. \Rightarrow \left\{ \begin{array}{l} {\rm{d}}u = - \frac{1}{{{x^2}}}{\rm{d}}x\\ v = f\left( x \right) \end{array} \right.\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 % da9maapehabaWaaeWaaeaadaWcaaqaaiqadAgagaqbamaabmaabaGa % amiEaaGaayjkaiaawMcaaiabgUcaRiaaikdaaeaacaWG4baaaiabgk % HiTmaalaaabaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiab % gUcaRiaaigdaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaaaaaOGaay % jkaiaawMcaaiaabsgacaWG4baaleaacaaIXaaabaGaaGOmaaqdcqGH % RiI8aaaa!4D39! J = \int\limits_1^2 {\left( {\frac{{f'\left( x \right) + 2}}{x} - \frac{{f\left( x \right) + 1}}{{{x^2}}}} \right){\rm{d}}x} \) \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaaq % GaaeaadaWcaaqaaiaaigdaaeaacaWG4baaaiaac6cacaWGMbWaaeWa % aeaacaWG4baacaGLOaGaayzkaaaacaGLiWoadaqhaaWcbaGaaGymaa % qaaiaaikdaaaGccqGHRaWkdaWdXbqaamaalaaabaGaamOzamaabmaa % baGaamiEaaGaayjkaiaawMcaaaqaaiaadIhadaahaaWcbeqaaiaaik % daaaaaaOGaaeizaiaadIhaaSqaaiaaigdaaeaacaaIYaaaniabgUIi % YdGccqGHsisldaWdXbqaamaalaaabaGaamOzamaabmaabaGaamiEaa % GaayjkaiaawMcaaaqaaiaadIhadaahaaWcbeqaaiaaikdaaaaaaOGa % aeizaiaadIhaaSqaaiaaigdaaeaacaaIYaaaniabgUIiYdGccqGHRa % WkdaWdXbqaamaabmaabaWaaSaaaeaacaaIYaaabaGaamiEaaaacqGH % sisldaWcaaqaaiaaigdaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaa % aaaOGaayjkaiaawMcaaiaabsgacaWG4baaleaacaaIXaaabaGaaGOm % aaqdcqGHRiI8aaaa!6616! = \left. {\frac{1}{x}.f\left( x \right)} \right|_1^2 + \int\limits_1^2 {\frac{{f\left( x \right)}}{{{x^2}}}{\rm{d}}x} - \int\limits_1^2 {\frac{{f\left( x \right)}}{{{x^2}}}{\rm{d}}x} + \int\limits_1^2 {\left( {\frac{2}{x} - \frac{1}{{{x^2}}}} \right){\rm{d}}x} \)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS % aaaeaacaaIXaaabaGaaGOmaaaacaWGMbWaaeWaaeaacaaIYaaacaGL % OaGaayzkaaGaeyOeI0IaamOzamaabmaabaGaaGymaaGaayjkaiaawM % caaiabgUcaRmaaeiaabaWaaeWaaeaacaaIYaGaciiBaiaac6gacaWG % 4bGaey4kaSYaaSaaaeaacaaIXaaabaGaamiEaaaaaiaawIcacaGLPa % aaaiaawIa7amaaDaaaleaacaaIXaaabaGaaGOmaaaakiabg2da9maa % laaabaGaaGymaaqaaiaaikdaaaGaey4kaSIaciiBaiaac6gacaaI0a % aaaa!51D4! = \frac{1}{2}f\left( 2 \right) - f\left( 1 \right) + \left. {\left( {2\ln x + \frac{1}{x}} \right)} \right|_1^2 = \frac{1}{2} + \ln 4\)