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根号? 根号! 换掉它! 根式代换 [092]
在这个视频中, 我们讲解了应用根式代换进行换元求积分. 在积分过程中, 如果遇上带有根式妨碍积分正常进行时, 我们可以考虑将根式作为一个整体进行换元, 从而简化积分难度.
高等数学(上)习题集81/264
![三角代换使用说明, cos^2t = 1 - sin^2t [091]](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/fc4b46ea8d6741a594ac5c46373d5fc3~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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=goyiFWP6V0deUDp8ChMA7Di%2BQpI%3D)
App11:49![根号? 根号! 换掉它! 根式代换 [092]](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/911fb0cbd92441f5a7134c6854171552~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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=3NPdbVx%2BVrQ%2BeRsqPf2sPlWTEvU%3D)
App09:58![根式代换就是替换掉不易求解的根号部分 [093]](http://p26-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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=anFt58cLKXltzgUX2LECWHj7rro%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://p26-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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=9LaVvwkfeDeVSGj0amo571UuNgw%3D)
App08:35![根式代换与三角代换都可以求解的两道题 [095]](http://p3-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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=Qp1BWLBi0kNzFeJB7M3i6E59%2Bjk%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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=kMAL6YPF5sKUhCRgl0VdiadN1AY%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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=yHjn3pdzoaWBLj543Hdgar3RrJQ%3D)
App06:05![凑微分嘛! 没有就构造[098]](http://p26-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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=t%2BUb34%2FjNK8J%2BCN8jqIeQWZrJwk%3D)
App04:23![一种常见的凑微分方式: 硬凑! 积分嘛, 不寒碜[099]](http://p26-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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=L7%2FYZU5zapuOIeo79uHeBhZxiTY%3D)
App04:58![[100]两种方法求解, 难度差很大](http://p26-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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=Xqj%2Bn7h0%2BldwehZRVRnHuJ%2BdlrU%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://p3-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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=7aYkmyQYAWRdR%2B0Gf8qEH1ThTkY%3D)
App03:51![[102]不用费曼积分法也能求.](http://p3-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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=C8jr7ZtVnZlmMlU4NUGdxaNF0nc%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://p3-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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=62Y%2FYrpddkn3Rh%2F8I%2FmXqWjW4lk%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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=8lC4ITs%2BKaPmlnSWJhyXjUhuC4o%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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=qJEMfRlmqgiV0JPDMTV8hAvRzuY%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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=SW2OwdvevPGKxKTNItbsHuK8Jhc%3D)
App02:49![[107] 利用已知函数的导数凑微分](http://p26-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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=lEfGDymekAtEeRr5fDAbZiVqWUI%3D)
App02:16![[108] 构造式地凑微分, 然后利用分部积分法求解](http://p3-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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=%2FQGKep1EDurhbcBRRLj21mJ5hHA%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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=y3upGKt5xMg57PmJJD3jhHClTMA%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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=R71pPqEG8v8S4b%2BSOSzaAfMCbFE%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://p26-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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=9ZWP3g9iVESc8zE35gmVCwq7NOc%3D)
App02:47![[117] 积化和差公式用于求积分](http://p26-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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=1FPx8Ztq%2F%2B%2BWNB9eNbnD5CTCEok%3D)
App02:20![[119] 不用万能代换, 构造式地凑微分](http://p3-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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=6pJT9ksI5IhUa0baHIy7lsY6moU%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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=QRt84v6JsC0TRXzOtcyCeYfGhAI%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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=HdjhanFqnIi%2BPlRqkVoOzDjIokM%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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=4FLfOaGER0mODZOcu4pl5Zxk%2FyA%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://p3-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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=egFvIxA2fcf2bmx9q%2FhO3egI%2BHA%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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=Te5h8216EemhggOUu0BiOmxKId8%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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=zsejzfchxIKrtgxTe7S4b5OlkOw%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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=53nzcV3ZjSkTpE%2BX3Mpf78bn1uE%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://p3-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=20250313210833D20E641E24A9456948D2&requestFrom=704&version_code=80402&x-expires=1773407314&x-signature=jICBUJl0kG7istgzEYRwROTmH1E%3D)
App05:01打开App观看
