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求积分技巧之分部积分法; 三指幂对反, 顺序不能忘[088]
在这个视频中, 我们介绍了通过分部积分法求解了 只有反三角函数或是对数函数的被积函数求积分时的解题思路.
高等数学(上)习题集78/264
![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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=Q5WOULTXoSyV5sSB0yk%2BoXYyWCU%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://p26-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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=0RZS93km2ZIWyOlQvzJh0sBUCEw%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=YH89t0cz8y46nL899lvjM6i7aNc%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=hnoyB%2F7k%2BEd2CDH16%2BGvGOC4JxI%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=JPp8cpJilPmJ7%2BIO44BZi9%2Fq9Wo%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=JV5evY5HqCr%2BOvv70xLcZT9cXIo%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=dEoT066s8%2BH7KFiXqAFRz9gyKEc%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=jJFKi5jj8yif0r794Y8iCvAgLX4%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=qBrkbPciat9Ri8jFehUTuvrlvu0%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=I5c%2FcKPGqceDOuFIRkINEg8lI8s%3D)
App06:05![凑微分嘛! 没有就构造[098]](http://p26-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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=P4FPh8Nxjx8qjhwC%2ByJMPw7PpuA%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=LvxBIVyNnurjY2B13r1EyGRWsEc%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=eHwbzzUhqlVS2DkD99VL3ECtCsE%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=bwEoq5XoZevDPR%2BnfL1P9o8blvw%3D)
App03:51![[102]不用费曼积分法也能求.](http://p26-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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=hyH%2BTapN6zqvkrst%2BjQE1IijMis%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=bM4DIOz6ZEaHLCsqEX%2F2cS0SnPk%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://p3-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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=eb4F68CtqY%2BYSkCfRLXJW7YaJqM%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=xPXArPwFIWFB6edBIEMXivjxFmE%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=lsRTXVV%2BPFYWnQUHJMfohashyPo%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=x69m1LP4oPWDKzWj%2Bl0x%2BysbksM%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=N%2Fs5tOJtWZ%2FnVOzivpAPQIRY0LY%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=aihR1u8B8o71evfDqiNRm%2Fib1Po%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=JCMI8eY9%2Ba1x7hh93yFzXhv3r5M%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&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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=aVQLSmMoPuMhdZ9jyL%2FQQqnUDgA%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&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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=8GLhWLi5uyDkYKH4k1FpCguJW0A%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=ReQE9YmthyStRFFs5NGjC44A95c%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=2j3ehWJfsCmFfYk6D7F0qU2kKK0%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=03zSoAh34ov4CHuJ%2BrCvaKo1w8k%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=1wi3YYQwVYtmjDWGqFh6yiRrHuI%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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=JBZwgxMukVW0bKDJyb7rEbS7Ql0%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://p26-sign.bdxiguaimg.com/tos-cn-i-0004/40e96956d2b940ce99a26b2c506f169a~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=20250313194016C0211516007147BB0CFB&requestFrom=704&version_code=80402&x-expires=1773402016&x-signature=Mu4%2FjCqYhxbJ4zRh9iWWIvmyXnk%3D)
App08:09
