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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]
在这个视频中, 我们介绍了通过裂项的技巧, 将被积函数转化为若干个可直接利用积分公式得到原函数的积分.
高等数学(上)习题集75/264
![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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=9qxD4NvgXJX4cj9OVTMX0TJll3c%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=sSvD5jDGknI8mkUOzWE1K9w7vn0%3D)
App14:06![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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=mzEk8lEmJC9f2XumSkrSaoaUFU8%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=DbSAZIHJSgJBjT%2BlFSLGXLtHEAw%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=nUwtEO%2FvFoXSbGHobGIiQ%2FGCuD4%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=G1%2FCC9nM0ojTogItOU30PkMRBkY%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=8MEY5Lf9CJdQIGjda18nzIcLvbc%3D)
App11:49![根号? 根号! 换掉它! 根式代换 [092]](http://p26-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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=9lNs6fpm%2F9s6DJyKP5e06Jm7KQk%3D)
App09:58![根式代换就是替换掉不易求解的根号部分 [093]](http://p26-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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=RERaAOlU1QLxIy6oRiBbK4EtA7k%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=UbNZCilCezGt0By80FBiQEF8%2FGI%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=%2Fz23d6183Sqe7nVcMxLlR3fFnFc%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=xjw5MV72%2Bb%2BP8bZ4rgv2vr9DUwI%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=O2gm02d6JcdeIdeJW6vxJfbEgEI%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=%2FdI6SjzR%2BPCew1kh7RJP3B%2F2MHI%3D)
App04:23![一种常见的凑微分方式: 硬凑! 积分嘛, 不寒碜[099]](http://p26-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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=10aQ90SvK7aTsdikj5tmBgCcIQg%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=58UO5wP1eWjC1n8WXiyhxlQh%2Fnk%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=Gerk7t034hI5UGxKHgFE4J9J2%2Bc%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=UbfJE8qQnJ%2BfNU8Y2aqXrd%2Bweq8%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=9Hlpu0Obj0SfGOFxDiKvxtnqBhg%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=tnyrH4imXE%2FVeMu2VT6toVgoUd4%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=Zbe3XxLQaemZSDvr7%2F6cqVmYbpA%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=pqJ3IL93wh%2B4hOmxKP89xz%2FC%2BoQ%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=WiU48TcsPBshHrqbEBzy7G%2BTRf0%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=t3V1peDiGYoJ%2BKphHksqYNg1Ods%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=QYiN1S5ku11iLuSJ7RNjx3wq9rE%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=LlrVCXcaVhF858R%2BD%2Fz0xRqGmTk%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=DS0Kt01d%2BIAdTZtz%2FFcLOtg2zHc%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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=lcYqU4T%2BAbrgq%2BCS6AXkbKVXzvo%3D)
App02:20![[119] 不用万能代换, 构造式地凑微分](http://p26-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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=jD26jV0%2FflMtHzdudgmDXxJVP1Y%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&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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=98zg%2Fu5URcfPPFxOvWLrhjR4HsU%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&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=20250313193124F6F7BDE89FDABACBDEF0&requestFrom=704&version_code=80402&x-expires=1773401485&x-signature=uboER3ukYnuMjvFZAEBBSxUJGIQ%3D)
App05:16
