Giá trị của tích phân \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2 % da9maapehabaGaamiEaiGacogacaGGVbGaai4CamaaCaaaleqabaGa % aGOmaaaakiaadIhacaqGKbGaamiEaaWcbaGaaGimaaqaamaalaaaba % GaeqiWdahabaGaaGOmaaaaa0Gaey4kIipaaaa!4517! I = \int\limits_0^{\frac{\pi }{2}} {x{{\cos }^2}x{\rm{d}}x} \) được biểu diễn dưới dạng \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaac6 % cacqaHapaCdaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGIbaaaa!3C05! a.{\pi ^2} + b\) \((a,b \in Q)\).Khi đó tích a.b bằng

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabe % qaaiaadwhacqGH9aqpcaWG4baabaGaaeizaabaaaaaaaaapeGaamOD % aiabg2da9iGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaaki % aadIhacaqGKbGaamiEaiabg2da9maalaaabaGaaGymaiabgUcaRiGa % cogacaGGVbGaai4CaiaaikdacaWG4baabaGaaGOmaaaacaqGKbGaam % iEaaaapaGaay5Eaaaaaa!4DAF! \left\{ \begin{array}{l} u = x\\ {\rm{d}}v = {\cos ^2}x{\rm{d}}x = \frac{{1 + \cos 2x}}{2}{\rm{d}}x \end{array} \right.\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qadaGabaabaeqabaGaaeizaiaadwhacqGH9aqpcaqGKbGaamODaaqa % aiaadAhacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiaadIhacq % GHRaWkdaWcaaqaaiaaigdaaeaacaaI0aaaaiGacohacaGGPbGaaiOB % aiaaykW7caaIYaGaamiEaaaacaGL7baaaaa!490A! \Rightarrow\left\{ \begin{array}{l} {\rm{d}}u = {\rm{d}}v\\ v = \frac{1}{2}x + \frac{1}{4}\sin \,2x \end{array} \right.\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2 % da9iaadIhadaqadaqaamaalaaabaGaaGymaaqaaiaaikdaaaGaamiE % aiabgUcaRmaalaaabaGaaGymaaqaaiaaisdaaaGaci4CaiaacMgaca % GGUbGaaGPaVlaaikdacaWG4baacaGLOaGaayzkaaWaaqqaaqaabeqa % amaalaaabaGaeqiWdahabaGaaGOmaaaaaeaacaaIWaaaaiaawEa7ai % abgkHiTmaapehabaWaaeWaaeaadaWcaaqaaiaaigdaaeaacaaIYaaa % aiaadIhacqGHRaWkdaWcaaqaaiaaigdaaeaacaaI0aaaaiGacohaca % GGPbGaaiOBaiaaykW7caaIYaGaamiEaaGaayjkaiaawMcaaiaadsga % caWG4baaleaacaaIWaaabaWaaSaaaeaacqaHapaCaeaacaaIYaaaaa % qdcqGHRiI8aaaa!5F49! I = x\left( {\frac{1}{2}x + \frac{1}{4}\sin \,2x} \right)\left| \begin{array}{l} \frac{\pi }{2}\\ 0 \end{array} \right. - \int\limits_0^{\frac{\pi }{2}} {\left( {\frac{1}{2}x + \frac{1}{4}\sin \,2x} \right)dx} \)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS % aaaeaacqaHapaCdaahaaWcbeqaaiaaikdaaaaakeaacaaI4aaaaiab % gkHiTmaabmaabaWaaSaaaeaacaaIXaaabaGaaGinaaaacaWG4bWaaW % baaSqabeaacaaIYaaaaOGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGio % aaaaciGGJbGaai4BaiaacohacaaIYaGaamiEaaGaayjkaiaawMcaam % aaeeaaeaqabeaadaWcaaqaaiabec8aWbqaaiaaikdaaaaabaGaaGim % aaaacaGLhWoaaaa!4C4E! = \frac{{{\pi ^2}}}{8} - \left( {\frac{1}{4}{x^2} - \frac{1}{8}\cos 2x} \right)\left| \begin{array}{l} \frac{\pi }{2}\\ 0 \end{array} \right.\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS % aaaeaacqaHapaCdaahaaWcbeqaaiaaikdaaaaakeaacaaI4aaaaiab % gkHiTmaabmaabaWaaSaaaeaacaaIXaaabaGaaGinaaaadaWcaaqaai % abec8aWnaaCaaaleqabaGaaGOmaaaaaOqaaiaaisdaaaGaeyOeI0Ya % aSaaaeaacaaIXaaabaGaaGioaaaadaqadaqaaiabgkHiTiaaigdacq % GHsislcaaIXaaacaGLOaGaayzkaaaacaGLOaGaayzkaaaaaa!494C! = \frac{{{\pi ^2}}}{8} - \left( {\frac{1}{4}\frac{{{\pi ^2}}}{4} - \frac{1}{8}\left( { - 1 - 1} \right)} \right)\) \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS % aaaeaacaaIXaaabaGaaGymaiaaiAdaaaGaeqiWda3aaWbaaSqabeaa % caaIYaaaaOGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGinaaaaaaa!3E66! = \frac{1}{{16}}{\pi ^2} - \frac{1}{4}\)

Theo giả thiết \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2 % da9iaadggacaGGUaGaeqiWda3aaWbaaSqabeaacaaIYaaaaOGaey4k % aSIaamOyaaaa!3DD9! I = a.{\pi ^2} + b\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabe % qaaiaadggacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIXaGaaGOnaaaa % aeaacaWGIbGaeyypa0JaeyOeI0YaaSaaaeaacaaIXaaabaGaaGinaa % aaaaGaay5Eaaaaaa!3FAA! \left\{ \begin{array}{l} a = \frac{1}{{16}}\\ b = - \frac{1}{4} \end{array} \right.\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taam % yyaiaac6cacaWGIbGaeyypa0JaeyOeI0YaaSaaaeaacaaIXaaabaGa % aGOnaiaaisdaaaaaaa!3F0C! \Rightarrow a.b = - \frac{1}{{64}}\)