Nguyên hàm F(x) của hàm số \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqaMrdvLHfij5gC1rhimfMBNvxyNvgaMXfBLzgDOa % cEGWLCPDgA0LspCzMCHn2EX03EYG3kX0hatCvAUfeBSjuyZL2yd9gz % LbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYL % wzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9 % v8qqaqFr0xc9pk0xbba9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0F % b9pgeaYRXxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaa % aOqaaabaaaaaaaaapeGaamOzamaabmaapaqaa8qacaWG4baacaGLOa % GaayzkaaGaeyypa0ZaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaikda % caWG4bGaey4kaSIaaGymaaaaaaa!5169! f\left( x \right) = \frac{1}{{2x + 1}}\), biết \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqaMrevLHfij5gC1rhimfMBNvxyNvgagXfBLzgDOa % cxMjxyJThx0vgE0Txz91sm9TNm9bcxYL2zOrxk9WLzYf2y7ntF7jtF % amXvP5wqSXMqHnxAJn0BKvguHDwzZbqefqvATv2CG4uz3bIuV1wyUb % qedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8Yj % Y-vipgYlh9vqqj-hEeeu0xXdbba9frFj0-OqFfea0dXdd9vqai-hGu % Q8kuc9pgc9q8qqaq-dir-f0-yqaiVgFr0xfr-xfr-xb9adbaqaaeGa % ciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacaWGgbWaaeWaa8 % aabaWdbmaalaaapaqaa8qacaqGLbGaeyOeI0IaaGymaaWdaeaapeGa % aGOmaaaaaiaawIcacaGLPaaacqGH9aqpdaWcaaWdaeaapeGaaG4maa % WdaeaapeGaaGOmaaaaaaa!57EA! F\left( {\frac{{{\rm{e}} - 1}}{2}} \right) = \frac{3}{2}\) là:

Áp dụng công thức nguyên hàm mở rộng

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqaMvevLHfij5gC1rhimfMBNvxyNvga7zexSvMz0H % ci4bcxYL2zOrxk9WvAUr3ECzMCHn2EX03EYG3kX0hx0vgE0Thz9HxF % 9bWexLMBbXgBcf2CPn2qVrwzqf2zLnharuavP1wzZbItLDhis9wBH5 % garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYl % Ni-xH8yiVC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbbG8Fa % sPYRqj0-yi0dXdbba9pGe9xq-JbbG8A8frFve9Fve9Ff0dmeaabaqa % ciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbiaadAeadaqada % WdaeaapeGaamiEaaGaayjkaiaawMcaaiabg2da9maapeaapaqaa8qa % daWcaaWdaeaapeGaaGymaaWdaeaapeGaaGOmaiaadIhacqGHRaWkca % aIXaaaaiaabsgacaWG4baaleqabeqdcqGHRiI8aaaa!5CC4! F\left( x \right) = \int {\frac{1}{{2x + 1}}{\rm{d}}x} \)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqaMTdvLHfij5gC1rhimfMBNvxyNvga71dxMjxyJT % xm9TNm9XfBUbcxSvMz0XhiYG3kXacxYL2zOrhFRmuFamXvP5wqSXMq % HnxAJn0BKvguHDwzZbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSb % qefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8YjY-vipgYlh9vq % qj-hEeeu0xXdbba9frFj0-OqFfea0dXdd9vqai-hGuQ8kuc9pgc9q8 % qqaq-dir-f0-yqaiVgFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqa % amaabaabaaGcbaaeaaaaaaaaa8qacqGH9aqpdaWcaaWdaeaapeGaaG % ymaaWdaeaapeGaaGOmaaaaciGGSbGaaiOBamaaemaapaqaa8qacaaI % YaGaamiEaiabgUcaRiaaigdaaiaawEa7caGLiWoacqGHRaWkcaWGdb % aaaa!5822! = \frac{1}{2}\ln \left| {2x + 1} \right| + C\)

Mà \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqaMrevLHfij5gC1rhimfMBNvxyNvgagXfBLzgDOa % cxMjxyJThx0vgE0Txz91sm9TNm9bcxYL2zOrxk9WLzYf2y7ntF7jtF % amXvP5wqSXMqHnxAJn0BKvguHDwzZbqefqvATv2CG4uz3bIuV1wyUb % qedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8Yj % Y-vipgYlh9vqqj-hEeeu0xXdbba9frFj0-OqFfea0dXdd9vqai-hGu % Q8kuc9pgc9q8qqaq-dir-f0-yqaiVgFr0xfr-xfr-xb9adbaqaaeGa % ciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacaWGgbWaaeWaa8 % aabaWdbmaalaaapaqaa8qacaqGLbGaeyOeI0IaaGymaaWdaeaapeGa % aGOmaaaaaiaawIcacaGLPaaacqGH9aqpdaWcaaWdaeaapeGaaG4maa % WdaeaapeGaaGOmaaaaaaa!57EA! F\left( {\frac{{{\rm{e}} - 1}}{2}} \right) = \frac{3}{2}\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqaMzhvLHfij5gC1rhimfMBNvxyNvgaCXuzMrNCPD % gA0fMCY92DGWLzYf2y7ftF7jtFCXMBGWfBLzgD8bImCXwzMrhkGWLz % Yf2y7XfDLHhD7vwFTetF7jtFGWLCPDgA0LYkXacxYL2zOrhFRmupCz % MCHn2EZ03EY0hatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhi % ov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4r % NCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9 % q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr0- % vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaapeGa % eyi1HS9aaSaaa8aabaWdbiaaigdaa8aabaWdbiaaikdaaaGaciiBai % aac6gadaabdaWdaeaapeGaaGOmamaabmaapaqaa8qadaWcaaWdaeaa % peGaaeyzaiabgkHiTiaaigdaa8aabaWdbiaaikdaaaaacaGLOaGaay % zkaaGaey4kaSIaaGymaaGaay5bSlaawIa7aiabgUcaRiaadoeacqGH % 9aqpdaWcaaWdaeaapeGaaG4maaWdaeaapeGaaGOmaaaaaaa!76FA! \Leftrightarrow \frac{1}{2}\ln \left| {2\left( {\frac{{{\rm{e}} - 1}}{2}} \right) + 1} \right| + C = \frac{3}{2}\)\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqaMDcvLHfij5gC1rhimfMBNvxyNvgaCXuzMrNCPD % gA0fMCY92DGmupXaWexLMBbXgBcf2CPn2qVrwzqf2zLnharuavP1wz % ZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGu % eE0jxyaibaieYlNi-xH8yiVC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vq % aqpepm0xbbG8FasPYRqj0-yi0dXdbba9pGe9xq-JbbG8A8frFve9Fv % e9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWd % biabgsDiBlaadoeacqGH9aqpcaaIXaaaaa!4901! \Leftrightarrow C = 1\)