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根式代换与三角代换都可以求解的两道题 [095]
高等数学(上)习题集84/264
![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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=r8RzyjKmQ3iysrlDZR8bEEkIopU%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=jg4SpIsWYY2GiTZ%2FfZ%2BGZvH8z7o%3D)
App06:08![三角代换能解的题, 我根式代换也想试一试[096]](http://p26-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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=BJ%2FS93eooY8UYqnOV5ZfMpz5hGU%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=uwQyY7cZV0OsoYzxsH5av3SXMC8%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=l%2B7jNp6MKBamJupgDnP0r5q90ZQ%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=4C4W7rKUkQSedF5fCAMoXeRczP8%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=UYtvLYptWCmQDyEtBoDo9ZCmmvI%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=aqMXmh7JrlGL9kE%2FHf0zcA2rN90%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=hdceOsrsq1ADIFQK%2BxtMzw6Wr%2Bg%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=d24q95sxAq4XBY7m%2BRx8PTDGzV0%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://p26-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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=24V5O6bQ2LBzDObLFXhpelb3OK8%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=UaITMvAqOh1eCBatr8XCHD9nRuU%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=Qjxf6fhXlIW0Xk%2BGBqESNmvt26A%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=S81scRRIfas6igQLbTf6OorPPAo%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=gz707qjeVE4w7rcEfBwNXRp5ySE%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=8hA4XQz52LPorxPi62xd7jP4IKw%3D)
App02:59![[111] 分部积分法+三角恒等式](http://p3-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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=qk7IWipheCBWv4PX0I1GJsO4l8s%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=Z%2FE9lWGVfHFmG246qpBNtJfNmKo%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=OJ5Ie1GrJ4Z6w8qE9PeAyt9Kh9U%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=xBTrRRkBbYhQCJKSBg4BmgBpwdQ%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=O52Ug9a%2Fv8iCGgRqrFDrWNwWaYY%3D)
App04:51![[高数122] 两次换元求解 | 先凑微分换元后三角代换](http://p26-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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=giieNlqMeHb%2F93Jd1VMwAMj21mM%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=WLgxhylB8v5%2FgL2fhvlvHoVWbTc%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=nERe2GpAHAva4Fn4S5KPagsdqCE%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://p26-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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=3j5QRuSDvi91YDRDMIs2jo5vLUY%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=EDyENh%2FT4fSxdNkmH3vcLB0SyFQ%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=uveZpdRu7ysiU%2FEGqZf1R64corQ%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=pKSm%2BNkpC7EbxRTziv2X9FBtDs4%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://p3-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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=qoumrMEfZup%2Fv7aFtrg8lIo7dSg%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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=%2BXGJr8i%2FOupWQebgyR8ICk40ZE8%3D)
App02:16![[高数131] 少见的换元方式|将难处理的函数看成一个整体](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/b39d02ebb23b4465b39a0a80f8ac32d7~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=20250313191023CA209D0D2CF8ABBB37C6&requestFrom=704&version_code=80402&x-expires=1773400223&x-signature=0cEhyNMRXLEDWshKwtTEJtB9Pf0%3D)
App05:23
