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微分的逆运算：从微分结果倒推原函数 | 微分的逆运算=不定积分
在这个视频中, 我们利用基本的微分公式, 从微分结果倒推原函数. 我们可以根据微分结果的样式, 结合基本微分公式, 获取原函数大致的样子; 然后将猜测函数进行微分运算, 两个结果进行比较, 确定二者的差异; 最后通过适当的调整(大部分情况是系数的调整)得到最终的答案. 一定要记得加上一个常数 C.
高等数学(上)习题集68/264
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App04:40![求积分技巧: 通过配方法,转化为基本积分表中的积分形式[084]](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/6d55da2b3be24bd19f56eaf69df5b898~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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=Io55oAqRsFWyL2Mm%2B7DV%2BMu5LHI%3D)
App10:05![求积分技巧之裂项求积,通过裂项转化为若干个积分公式的积分[085]](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/1388b6ea9efc470bbd77cf0b2bd20309~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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=ht5fEHZ4gYYrFbbOgJFhcdun3Qk%3D)
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App08:43![求积分技巧之凑微分法(2), 最基础也是最重要的求积分方法[087]](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/9de6447729cb4290a20668a0693a96a6~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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=O%2FMtrDFbG593SOzPkB0LVkw8XPA%3D)
App07:58![求积分技巧之分部积分法; 三指幂对反, 顺序不能忘[088]](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/b2cd6ade410a4c14b8075e354d01110c~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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=eTPI8YL33htCydQEpEXUoJbeIDU%3D)
App08:18![三角代换使用说明, sec^2t = 1 + tan^2t [090]](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/799c75786a0044b5958adf7d2ba7ff48~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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=qUgOB1baP0ajFJyIPVgB2OGtVKM%3D)
App09:27![三角代换使用说明, cos^2t = 1 - sin^2t [091]](http://p26-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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=VeBh9FlE%2B0dSXEg%2FmCPbbafMdUA%3D)
App11:49![根号? 根号! 换掉它! 根式代换 [092]](http://p3-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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=mCXjQ%2F0x3zjw9LPwULxCnogkjmE%3D)
App09:58![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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=IopT8HbjCgrW0%2FsE%2F%2BHH%2BTApjn4%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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=feNQc5hDp2SUV7Iqgp%2B4hwRgS6M%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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=udm4ce0tU2Msbhea5%2B2XpNS%2B%2BsQ%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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=nvM7ped736ziULbPSj8Q3dIhFhk%3D)
App05:21![好吧, 这里居然是用分部积分[097]](http://p26-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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=oeZX7eRxa6FSfQ99u97QcTtUsGk%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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=oDxeaspHcZNNRzPCZZykaPBu78k%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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=jHdEUQuzr%2BTh%2FS0LZWAoflm0zZo%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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=vHdaY18WiD80NEfanqww89Lu3Kk%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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=yBA1p2TSkdWOqWwb9A6HxmOhWAs%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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=is5S6J0rAmCDT2xzQcgWLdyWP%2Fg%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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=ljMpVUh2akyeEUWxd6USFGH2QPU%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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=Q8KaQ7DU9bpjMLfsgKRK4dwoRm4%3D)
App11:09![[105] 想凑微分? 得先熟悉函数的导数.](http://p26-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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=KLsOu3sbllkSm798e77NuMy3LJE%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&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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=7FuxrtZIGh5g7DDihxTfxdNNdX0%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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=bYUlr4znDUSX%2FeJESvVCehqI6WQ%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=2025031319003985472663371EC75329DD&requestFrom=704&version_code=80402&x-expires=1773399639&x-signature=mQkksFW%2BmnsP%2FlKdZmp%2F%2B2rtfdg%3D)
App03:17
