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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]
在这个视频中, 我们介绍了通过凑微分法求积分 这是求积分中最基础也是最重要的方法, 需要我们熟练掌握微分公式和积分公式. 从被积函数中找出需要凑微分的部分, 并使得凑完微分后的积分形式是易于求解的.
高等数学(上)习题集77/264
![求积分技巧之凑微分法(1), 最基础也是最重要的求积分方法[086]](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/8d8cdf6f5547400291dd60547e32a911~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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=E4Ba%2FMRsqlP435X%2FwNXIM4J5WaQ%3D)
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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=DnEeiUb%2BnslDklNAb7an4PVsJ3Q%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=FoRpo7erzxCEwtwX9zQ5YBBU%2BUk%3D)
App08:18![三角代换使用说明, sec^2t = 1 + tan^2t [090]](http://p3-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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=nkdsEtMgsMl3dnG6g3UmEVMU7FM%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=Bx4xwX1gYpzDOcyErJxvE%2FUjMwg%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=p6Ewmm%2B%2F6qrWY7XGbN0OIpAPDBU%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=wJGg9pYCNy0ELRbEVhKMexXEXgg%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=L1gN4Vsl40bpkUQRtTMaI8%2FOycE%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=fRLiM0onK0GlRaPVMpCbapNJsHg%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=P0lOnhxQIs4Mw2axNf8FVuEHVNw%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=w2ZNg9E6UJ%2FEeu043mBrmFFS2Y0%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=I9V%2BV0QuxyUUJpKMj%2BGclR0mMKY%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=Dx9yZwIwQa5SfXQ64jk8Kq8JnQQ%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=wFcPbOHAdAuZVJMCAwn6jWA25U0%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=LAXAsROH0kAZeh860Cl43TvaUFw%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=Le3MPf1CJ26seMMMD%2BNa8MzMizc%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=4JldD7jzFnIDVj3vYnmpQdUvx9g%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=xadF1kpTW4V%2FjPK2WHeshqFHJrc%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=Csx%2FBysKzO%2B93GHl67dtpouf4fw%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=3TQWrmIg6R7F6Y%2BI7stXS05JcSs%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=WrS%2FrZX7N%2B1twog26U0EBKMnB2I%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&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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=GdUVjapYwmzJCkGOOKB8G3fWn4U%3D)
App03:17![[109] 半角代换求不定积分](http://p26-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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=Lx6VTV%2Byg5z7Lvm2UaX8OQjw0Cc%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=DZfLM6gHtOqaqxduEg6CdWADNg0%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=pPMGjQi%2BSOvR79zowDACoUp%2B5Io%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=kZbIGaDRibb5uDEhlrKrGVGRKKw%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=d%2Fw5XcYPd%2BEzUZWUeYMGNw%2Fd%2FNI%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=ukKfZH%2B5z2W8XDwKd0r9VmpCTe0%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=wHBPN499D%2Bv6lCVgo7sVxxVaX3M%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=i2PrTBSlwUJcE1UZZlY4gCx0c0E%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=20250313202917E2274813C8913CBFBC71&requestFrom=704&version_code=80402&x-expires=1773404958&x-signature=rqb34V0p2Gc1FKHtOoCHz2vkWZM%3D)
App05:49
