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一种常见的凑微分方式: 硬凑! 积分嘛, 不寒碜[099]
在这个视频中, 我们通过构造分母的微分形式, 将分子拆成两部分, 第1部分正好是分母的微分; 第2部分只是一个常数1 进而, 将积分拆成两个易于求解的积分.
高等数学(上)习题集88/264
![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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=pIqYgf0vwKf5kwEny9kCUzks0zg%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=UmkmCyDy%2Bw4L9Irosw3gZIQir5w%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=UnBNt04SipMAIOktKQ78FCHkOr8%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=Ijjup7Sbzujxaexc49ffYqId8Tw%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=V87N3s4igshUiyzaYerrZDu3Xuk%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=KW%2B2Z32J8PNEKRCGGlCV0xY%2BuGI%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=eGGD7eYMqia2P0%2B%2BCMR7sAQEQCA%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=ahfZly7%2FViVTxRquLMvlA%2FSrwiE%3D)
App03:35![[106] 分部积分法 | 少见的凑微分方式](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/392b1829fd2f49f098ebf285347a9554~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=VYlcvOHE10F8ORIRM7ssf4F210w%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=841tJ9QyTeZAT4A6abaFHBgWUhY%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=BJcy88XibFHTAPRPMRdYeXN4bZw%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=L5XPcFZG6mb2bPWC6mD7ntCfeRA%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=VMewfOjVqLTdRYJth5ttWeDAhLI%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=8vsbniz8n7W4KXs0t0K0aluaJKk%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=%2F7UOFvepZTjs205b9IPzvxnUyuQ%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=B8DJZ26kjKRCz6qybJWGchbLItU%3D)
App03:46![[高数121] 三角代换求不定积分 | 思路直接且清晰](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/363005e55cf7440f96cd9ddf11a22efa~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=L9PM83PHGTy8Ry0NClkvBBDtmSE%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=aLuxCbpXYBBQdbZ0XtW5yhRzcYU%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=THY2v3Ky2NtoK86phqP33jAPxL4%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=izWcXqG6dP8KGBD0FeNJYSC0rv8%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=%2FzIQeo3AXCiPk30BHTpePwHzd0I%3D)
App08:09![[高数126] 三角代换|令x=sint,注意t的取值范围](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/7e1af70db9eb4f86a7feb78282adcab7~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=Ta5fAjisSP4RY9qMXX7tNL%2BtauY%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://p26-sign.bdxiguaimg.com/tos-cn-i-0004/5f6adb68603b44b1b154b7770677be6c~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=3wbPrxamsqdZxvQggc7319kDZuA%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=OTL5LNAJqFg%2FCropBs04iFDH1yQ%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=AnOCaLI%2FM0EOeTH6rYgdHddxFN8%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=NI4G%2FFaRbjMiFsLE4bQgWMgYHoE%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=wFlKgsK6dESh2HxYo1p5BCMG0Xc%3D)
App05:23![[高数132] 少见的换元方式|换元后分部积分法求解](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/a0d0cb73da794ac0b7aded29d90e4e57~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=kdJp4D3%2Bq7dO3%2BrqOV4g%2F7b0yoc%3D)
App03:41![[高数133] 有理函数积分|待定系数法拆项](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/943dcaf5cc86498cae6d100880909a4a~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=V19DDSvMgq2nH%2FBSr4YrmMu%2BLtw%3D)
App06:00![[高数134] 根据题意先求被积函数的表达式再求不定积分](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/9e80d5bb6f1c4d019daebe9add9558bf~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=2%2BBt%2BQStg9iVU%2Biwnv3tu5t5OgA%3D)
App04:35![[高数135] 不定积分等式两边同时求导得到f(x)的表达式](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/b6e002f18ba948869837965c0878cf2b~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075349CA3A9CD1200F3B61A984&requestFrom=704&version_code=80402&x-expires=1762386830&x-signature=D6PBaW1b%2BAj777%2BGBwOuKSZTNYI%3D)
App02:52打开App观看
