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两道解题思路不拐弯的积分题|第二类换元法|裂项求积 [094]
在这个视频中, 我们分别用三角代换和换元法求解这两道积分. 它们的思路是比较直接的. 求积过程中还使用了加项减项, 裂项求积等小技巧.
高等数学(上)习题集83/264
![根式代换就是替换掉不易求解的根号部分 [093]](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/2f5944aa6d734c3d96a519c34f127c88~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=QI%2FaEUcc%2FXRXpNoCy7EbJfDVCOQ%3D)
App09:06![两道解题思路不拐弯的积分题|第二类换元法|裂项求积 [094]](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/51fe2817a9f14283b33219e7901dc016~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=mvifAY4mDeUen9U7iLitepObZk0%3D)
App08:35![根式代换与三角代换都可以求解的两道题 [095]](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/f040bca229ed4f94a33f87c1f89d2ceb~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=XmbK3BGIg1cqwpFXc1bnhKi2kOg%3D)
App06:08![三角代换能解的题, 我根式代换也想试一试[096]](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/200afc81ad974f40be25d1e14e04a8c2~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=lQuG2CQ5ii5RTiRH8gxYDenCQt0%3D)
App05:21![好吧, 这里居然是用分部积分[097]](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/8a61e746eb044856b425d15b515eb765~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=B78MLAtOiNsH%2FJo9o%2FYXl8V7W%2Bs%3D)
App06:05![凑微分嘛! 没有就构造[098]](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/34bbb99701004974983f85acc5b98194~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=du72Pp1JoH3e8xjB%2FxPGJcDHftg%3D)
App04:23![一种常见的凑微分方式: 硬凑! 积分嘛, 不寒碜[099]](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/9008b820b68e44d5b6217aebbee651bd~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=9RlDXfMIRmMYKrPVId8ngEjaROg%3D)
App04:58![[100]两种方法求解, 难度差很大](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/7d2f088e942848438d530894bda7571d~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=pvAOlVjaznfX1m75HDuqHOOJo1o%3D)
App14:01![[101]有趣的换元方式; 拆项后将某一部分积分转化为所求积分](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/4f81bf6fb9ba4d1f8280c04bf57c2b7c~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=%2Bzm%2FNlwJCkivv%2BtnGAUQjSxgLU0%3D)
App03:51![[102]不用费曼积分法也能求.](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/edac18a3965a4d478b0fa1d4a22d656f~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=7AAi38OWIPUGEZ4Dk93SOWOvCgE%3D)
App05:18![[103]两种方法求解这道不定积分. 万能代换|三角变换](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/eda1c5848484405d92f9ac9a0840fe06~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=j9u9EJyyyhbijS0fjlDIQFBXbuE%3D)
App08:13![[104]两种方法半角代换|三角变换; 技巧性比较强的两种解法](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/d8a21c8c4e6d4dde8728b5ed7e93dae6~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=7aFFkF20yh2dZCFkT85nFLP9uFk%3D)
App11:09![[105] 想凑微分? 得先熟悉函数的导数.](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/8e2d23f4de434840a1b67605879ffc0e~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=4Fa80JUUsXa5RbVC5IHiVeGSWR8%3D)
App03:35![[106] 分部积分法 | 少见的凑微分方式](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/392b1829fd2f49f098ebf285347a9554~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=KY%2FqVAlCOPH92xg6qW1rvY8LgTU%3D)
App02:49![[107] 利用已知函数的导数凑微分](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/4918ebba85414bc086c4fc1ee73c0f0c~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=n210XW%2BULk4EwkKcGWZO4pU%2BBUs%3D)
App02:16![[108] 构造式地凑微分, 然后利用分部积分法求解](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/276ceae24da64de99525427550fda7d7~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=o6qLM9XmJeDtxDyNmzqQSNBW67w%3D)
App03:17![[109] 半角代换求不定积分](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/0292a09e40964315adb611437e1d68cd~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=LRh2h7U%2FMU7OQkVTMvXSc22G2sI%3D)
App02:59![[111] 分部积分法+三角恒等式](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/f3caaa4dce0c43deb5b550bbad82b1b9~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=vzwr%2BUqOc8ofP4COUwIHHiF1WbU%3D)
App03:07![[115] 三角变换多种多样. 这里先凑微分,后换元, 再拆项求积](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/0eb6421bf94e49c4ba50dc4260af525b~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=rLeZc4ptBbeuLSLhlgrLayFU7XA%3D)
App02:47![[117] 积化和差公式用于求积分](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/d47db3a5891e4005b8d083659df96c63~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=hHGFElSX1KnvObdtknsI8w7ulfk%3D)
App02:20![[119] 不用万能代换, 构造式地凑微分](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/bff57a998ab343bf9e80015a32eef678~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=8v1cRUHH4LSkjZkG%2FTxL6Ue0%2F40%3D)
App03:46![[高数121] 三角代换求不定积分 | 思路直接且清晰](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/363005e55cf7440f96cd9ddf11a22efa~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=JXNgxEuLO%2B30WcULlgISCYBcOc4%3D)
App04:51![[高数122] 两次换元求解 | 先凑微分换元后三角代换](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/82229330f4234c9b8655fbf711d238ba~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=bs1Z%2Fhn4RU4hy%2FCU654Ad5usA5I%3D)
App05:16![[高数123] 倒代换求不定积分 | 1/x 难处理,那就视为整体替换掉.](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/52aa9d26e9a54f388cb1b70df97997fb~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=%2FvM1gyXJAuWRGu1LrHUE0FivS9M%3D)
App03:01![[高数124] 典型的三角代换思路|换元后再利用分部积分求解](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/63532e3d3d504ff295d532dade0b04eb~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=2DX9qx0iwccvvkMyE3nC8RjoWd8%3D)
App05:49![[高数125] 灵活的三角代换|根据被积函数的特点令x=cost](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/40e96956d2b940ce99a26b2c506f169a~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=ROOhk5Zq2gi%2BgMwllPoiSV4p2TY%3D)
App08:09![[高数126] 三角代换|令x=sint,注意t的取值范围](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/7e1af70db9eb4f86a7feb78282adcab7~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=OV%2FyQzkjula8GSK0aosr15MVy70%3D)
App04:40![[高数127] 三角代换|令x=tant|换元后转化为分部积分法求解](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/5f6adb68603b44b1b154b7770677be6c~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=O0JLXRfCfTtdYlANICXko3W2%2BzE%3D)
App04:52![[高数128] 常用的e^x凑微分技巧|凑微分|分部积分|三角代换求解](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/00f854bb193641bf98747c51a5c977c6~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=Q4DKEeY0I7e83Yf6E6HJPRlBln0%3D)
App05:01![[高数129]第二类换元法-难处理的部分看成一个整体](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/c5f162c4af7f4816933457eb22d22aa1~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=%2FaEWnSH4cII0Gk5QQt%2FbMNc2fyE%3D)
App05:39![[高数130]利用三角恒等式进行适当变换后分部积分求解](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/15ccd9c78b084386863964c1240e09cc~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&biz_tag=list_custom%2Fnone&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20250313191108A73BE5B7E8BE3CBD5102&requestFrom=704&version_code=80402&x-expires=1773400269&x-signature=ieNYNirQUMNVCKwViwXJVCqCfMg%3D)
App02:16
