Cho \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaaca % WG4bGaamyzamaaCaaaleqabaGaaGOmaiaadIhaaaGccaqGKbGaamiE % aaWcbaGaaGimaaqaaiaaigdaa0Gaey4kIipakiabg2da9iaadggaca % WGLbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyaaaa!4528! \int\limits_0^1 {x{e^{2x}}{\rm{d}}x} = a{e^2} + b\); \((a,b \in Q)\). Tính a+ b

Giả sử \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2 % da9maapehabaGaamiEaiaadwgadaahaaWcbeqaaiaaikdacaWG4baa % aOGaaeizaiaadIhaaSqaaiaaicdaaeaacaaIXaaaniabgUIiYdaaaa!4160! I = \int\limits_0^1 {x{e^{2x}}{\rm{d}}x} \)

Đặt \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabe % qaaiaadwhacqGH9aqpcaWG4baabaGaaeizaiaadAhacqGH9aqpcaWG % LbWaaWbaaSqabeaacaaIYaGaamiEaaaakiaabsgacaWG4baaaiaawU % haaiabgkDiEpaaceaaeaqabeaacaqGKbGaamyDaiabg2da9iaabsga % caWG4baabaGaamODaiabg2da9maalaaabaGaaGymaaqaaiaaikdaaa % GaamyzamaaCaaaleqabaGaaGOmaiaadIhaaaaaaOGaay5Eaaaaaa!5063! \left\{ \begin{array}{l} u = x\\ {\rm{d}}v = {e^{2x}}{\rm{d}}x \end{array} \right. \Rightarrow \left\{ \begin{array}{l} {\rm{d}}u = {\rm{d}}x\\ v = \frac{1}{2}{e^{2x}} \end{array} \right.\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taam % ysaiabg2da9maalaaabaGaamiEaaqaaiaaikdaaaGaamyzamaaCaaa % leqabaGaaGOmaiaadIhaaaGcdaabbaabaeqabaGaaGymaaqaaiaaic % daaaGaay5bSdGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaadaWd % XbqaaiaadwgadaahaaWcbeqaaiaaikdacaWG4baaaOGaaeizaiaadI % haaSqaaiaaicdaaeaacaaIXaaaniabgUIiYdGccqGH9aqpdaWcaaqa % aiaadwgadaahaaWcbeqaaiaaikdaaaaakeaacaaIYaaaaiabgkHiTm % aalaaabaGaaGymaaqaaiaaisdaaaGaamyzamaaCaaaleqabaGaaGOm % aiaadIhaaaGcdaabbaabaeqabaGaaGymaaqaaiaaicdaaaGaay5bSd % Gaeyypa0ZaaSaaaeaacaWGLbWaaWbaaSqabeaacaaIYaaaaaGcbaGa % aGinaaaacqGHRaWkdaWcaaqaaiaaigdaaeaacaaI0aaaaaaa!5F1D! \Rightarrow I = \frac{x}{2}{e^{2x}}\left| \begin{array}{l} 1\\ 0 \end{array} \right. - \frac{1}{2}\int\limits_0^1 {{e^{2x}}{\rm{d}}x} = \frac{{{e^2}}}{2} - \frac{1}{4}{e^{2x}}\left| \begin{array}{l} 1\\ 0 \end{array} \right. = \frac{{{e^2}}}{4} + \frac{1}{4}\)

Từ giả thiết suy ra  \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2 % da9iaadkgacqGH9aqpdaWcaaqaaiaaigdaaeaacaaI0aaaaaaa!3B56! a = b = \frac{1}{4}\). Nên _{\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgU % caRiaadkgacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaaaa!3B30! a + b = \frac{1}{2}\)}