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一种常见的凑微分方式: 硬凑! 积分嘛, 不寒碜[099]
在这个视频中, 我们通过构造分母的微分形式, 将分子拆成两部分, 第1部分正好是分母的微分; 第2部分只是一个常数1 进而, 将积分拆成两个易于求解的积分.
高等数学(上)习题集88/264
![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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=3kslM53gD90LKrjVJIhw5rfsp5o%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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=Wy5DGJJURjUdOZMluriqZSUi7Mg%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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=8mf1DvxHj7kHEZPBQNVJ4UPeUzY%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://p3-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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=NppXrV1KT3fz16%2BUGqY4Vd8y2c0%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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=teXvbHRjLKf4qpKV4LuOIg4eU80%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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=s%2F%2BlCa2BuZDLZtd2OETvuthRPB4%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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=6NIPUdlhYcZlPUhBWe2kYQz7T18%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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=N4tHPNmbPZIO1F7gyzu%2FnhT8Ox8%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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=J37rbja%2BjVH9GezOe4IsUiIo%2Fcg%3D)
App02:49![[107] 利用已知函数的导数凑微分](http://p3-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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=7Dye1uV8%2F6jby11JgPrLzoS2dvk%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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=s1EGr5e352sXo5lHNp3T3X2kiUY%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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=xO69s6qRTjmfsaCXcWZaETuPnvA%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&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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=fquJdaNzlhoam0%2BmqtK30%2F236jg%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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=ji1wJywy032jCsffFKe5Smf1t1E%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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=0ZBmUvnJdBzsuOn%2F3cMvQwqZ3cE%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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=8l0N33kg3wdeMgLN7fUodwsJc8c%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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=c8ReS1SlgLTsxlkGDSUPylGnGGs%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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=oyfnEwL56xVdh3KC%2FwNTBdR7lVM%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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=Bxe%2BRpH5nPCuyr36M4%2FZRs6wTvk%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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=z2l96V9R1MK9Tr2HS2VORmjwhXw%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&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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=LGv4loAzLl%2BXQvdOu%2F2jURMHd3o%3D)
App08:09![[高数126] 三角代换|令x=sint,注意t的取值范围](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/7e1af70db9eb4f86a7feb78282adcab7~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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=VN41z%2B%2BJOHNFw70ksdLzWFcbsLw%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://p26-sign.bdxiguaimg.com/tos-cn-i-0004/5f6adb68603b44b1b154b7770677be6c~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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=gQikOLjQPLMkYC69ubcEdwBy55o%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&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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=hkK0YlUQ%2BV4jnzQDrBMT7Yhcd14%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://p3-sign.bdxiguaimg.com/tos-cn-i-0004/c5f162c4af7f4816933457eb22d22aa1~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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=r5tVArHNojqQgvmJHPtBHituOhM%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://p26-sign.bdxiguaimg.com/tos-cn-i-0004/15ccd9c78b084386863964c1240e09cc~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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=SLKf7zdk%2BGn%2FC1OFPrgLcBQ1ERI%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://p3-sign.bdxiguaimg.com/tos-cn-i-0004/b39d02ebb23b4465b39a0a80f8ac32d7~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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=l6r20xZKPBAnCiXC3SUkDz3dtbs%3D)
App05:23![[高数132] 少见的换元方式|换元后分部积分法求解](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/a0d0cb73da794ac0b7aded29d90e4e57~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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=OFuqW9YBeVRv%2BX1AqyEHOCtcL%2Fo%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&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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=2j2qaxmjYtbzL0Jdn7JhxO0K4dQ%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://p3-sign.bdxiguaimg.com/tos-cn-i-0004/9e80d5bb6f1c4d019daebe9add9558bf~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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=2twvjXgfmmSu2PWuymYtbWDs6c4%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://p26-sign.bdxiguaimg.com/tos-cn-i-0004/b6e002f18ba948869837965c0878cf2b~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=202503131904403934FF2A4B6EED8E1658&requestFrom=704&version_code=80402&x-expires=1773399881&x-signature=oGi%2BQsrzm3VxM88cxAGWc1ZDWis%3D)
App02:52
