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[109] 半角代换求不定积分
在这个视频中, 我们按以下几个步骤完成不定积分 1. 利用半角公式将被积函数转化为两个易于求解的函数 2. 利用分部积分法求第1部分积分 3. 得到积分与第2部分积分抵消, 最后得到解.
高等数学(上)习题集98/264
![[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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=nqRqQmFjG73G9T7KjBLLP7px2JU%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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=EaK9lA1N3vNaRvRySg2xSEiRWlI%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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=5y0nqV%2BvniXW9P9NFcFCMoRtkcY%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=8aQCoSprVLjn%2FKy%2FHNgA%2FH2%2Bm5I%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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=a51dKCXxbuNh%2BO2IsbyGQWLTZQE%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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=fMIVmwiNBFWyxIewQnxQTouZPBY%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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=jZQfvao8vXTr4UdO7v7GCHHdUag%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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=gjkQVVKAX4ikfjCppSecAIK8TeQ%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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=UUEaW1l1zcfn86msgLiztjf1DbE%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=4g6tR4OjtNtTy3LPoIEO%2F%2Bid2Kk%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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=QiUgucvMETzlMDl8ncMuW%2BoGHI0%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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=LnzmRj%2FTs1ZdLbfNrwSPGIlYDTA%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=JYL3Ko8NqUo8wneJ%2FtTFx5gOvAk%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=OmFcCbM0%2B5sLFISSWID5xIKxn%2B8%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=Vywp8wjCpRBOLbLPuDe6RxRKHWw%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://p3-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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=txpFhQgcbTPa4i9ASDINTxMw0y8%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://p26-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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=pkvAS6cOdGSzJpI3UuViNkvR0QI%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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=q6IROKVW0zlbaA5ehORSlCin7DA%3D)
App03:41![[高数133] 有理函数积分|待定系数法拆项](http://p3-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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=rpN5SAkLNP%2BqINTUmZ2QfGFH8YI%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://p26-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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=YFkqJ7UkOFjeZ2l%2FJmXPKBOejCQ%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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=HNVJewU%2BsEHNa4ZJ5QUhwVi4Yso%3D)
App02:52![[高数136] 先求表达式|后求不定积分|换元法+分部积分法](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/c1688590d82748d7835fe647e74eb1d5~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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=S3NQ1Nvh13ght2Dp6AjhaH5uc1Y%3D)
App06:36![[高数137] 挺简单的一道题|前提是记得cosx的任意阶导数公式](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/131053a91bc647fd98236c8fbda23f74~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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=7%2BKslB3xYUi726XInpXT7BptdLw%3D)
App01:51![[高数138] 三角恒等式+分部积分法得递推公式](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/f15b85149a964e0a83eb81d9e3c8dbcc~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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=o8lvf1ThsXCFqDO82DnSQGX08IY%3D)
App02:29![[高数139] 裂项求积分+分部积分法得递推公式](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/7fd9d02f10684e46a8f54e1c4728a075~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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=FHg8I5WIVHz1JaPK3vnPtLvsZ9E%3D)
App04:55![[高数140] 三角变换+分部积分法得递推公式](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/b383e1867cc14120928576bfdb115ed2~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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=0CexhruXdhu%2BzmGVfHcwrqdKvm0%3D)
App04:45![[高数141]求极限问题|利用等价无穷小替换和常见的极限](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/3c63f99ea0e3441eb9c8fbdf1fc9070a~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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=xKpSyQAEIUHErGlScnrJhnvxtlo%3D)
App05:11![[高数142]求极限问题|三种方法求无穷个无穷小量的和的极限](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/572c41cb33904509866bc34f52c7d29e~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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=F5wytc1%2B%2FSyU2tkhkz86t31Qyc8%3D)
App12:06![[高数143]求极限问题|两数列乘积的均值的极限为两数列极限的乘积](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/f84cb6d5302d449b8dd9571bad01b44c~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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=MPWqkporIHzTmhkITs9iOp7uig8%3D)
App08:45![[高数144]求极限|利用欧拉常数相关极限和Stirling公式](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/ecdd2e9f0f8f426fb1d02d57c4648262~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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=8O2DA%2BWdR%2BkKa0ZW%2F9Eyy4MObhE%3D)
App09:18![[高数145]求极限|取整函数相关的极限问题](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/3934a7553c934b69acdb94d4fa4db1a9~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=202411060754143D14AE37732D1B57EC8B&requestFrom=704&version_code=80402&x-expires=1762386854&x-signature=cTqaX8FZX3Rars8zgLGZn6Fr2FE%3D)
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