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三角代换使用说明, cos^2t = 1 - sin^2t [091]
在这个视频中, 我们讲解了应用三角代换进行换元求积分. 主要是使用了三角恒等式中的 cos^2t = 1 - sin^2t 换元时, 主要是观察到被积函数包含了根号下的二次多项多, 而且无法直接应用积分公式求解.
高等数学(上)习题集80/264
![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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=UOazk%2Fw9FmUbHF0rxoDRINHbpv8%3D)
App09:27![三角代换使用说明, cos^2t = 1 - sin^2t [091]](http://p3-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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=lFJD65lxvhvdAScfJB323pnqRvQ%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=ZiYWCfQ8vBndEOmrg%2Fb7YKfiDpY%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=ruC0oUwK7JQ8u01MlI58q9iO3r4%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=1DK84Nj5Jl7BBWn3JMkTqulkJSA%3D)
App08:35![根式代换与三角代换都可以求解的两道题 [095]](http://p26-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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=B9REM6diQxIrfbJuS2AQv3STUA0%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=DAfV23dk31saj6xjSMUvIrF2aNU%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=CHVQ%2FYxdFd6kONkJ2D%2FL8sbavD0%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=jRG4J9TG6BuR%2B8Auq8ky7BFwL%2FA%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=qMkwQVU1BOEKK792D%2FMt1h65jn0%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=CZ9i6wHxWkRomgi%2Bwg38Mhx2Soc%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=jp6D8l695IUjQRB7QrtFZzfMxqA%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=V8Q4Ged%2Bdh4fXMWyOQzk5xkIZ5U%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=ruC15o1FvQZxee9dhICMmIA0zko%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=0VfWyDNQ%2Bh32M59V%2F3hnXVLr%2FV8%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=b6L5jzEwzO9PcgzUPpcels4mKuA%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=OFjBeyt3%2B%2BYHFmDU3sd88iB475U%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=gu%2BW6PQoINHvfp%2FQJAT7HgoFpvQ%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=d7AmPCWuG7j77zdvkkepOTN1nss%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=F%2Bt8HzlnW0JXOsB86t3%2Bqug86%2BE%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=idXWoLTkTQDBEXAA7pAIrWjrPLc%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=WDnVLheC2RmbRo3yKbGSCp%2BsQBo%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=CkRaSXH1bSs0n7OBYUQ3VyZcWcw%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=EcEjIoZrSBzk2ptPVksVBGh%2F1KM%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=NgIzUiMdm4JzjejAFHFzYxEhFjw%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=XD%2Bii6r7Xi5CgGSdHS8Lod%2Brfx4%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=RnqdGDluB1ma62dEcA6zGS%2FDjeA%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=POJp0K2pw6J4YwLV03D8gpFU%2Fjk%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=9mKAAfiebT8l3GN3fDRGKuknPwQ%3D)
App08:09![[高数126] 三角代换|令x=sint,注意t的取值范围](http://p26-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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=74NQK6oaCR1LKxn2lr8lmy3OoyU%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=202503132046384FEA72855AB5F56564CA&requestFrom=704&version_code=80402&x-expires=1773405998&x-signature=SCJ5DREtJOTsOyKrj5X8RN1WI%2Bw%3D)
App04:52
