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[高数140] 三角变换+分部积分法得递推公式
高等数学习题讲解 第140题 适当的三角变+分部积分法得递推公式
高等数学(上)习题集122/264
![[高数139] 裂项求积分+分部积分法得递推公式](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/7fd9d02f10684e46a8f54e1c4728a075~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=zDb9ZNP6rFryMY4Ktmg3LJASz2A%3D)
App04:55![[高数140] 三角变换+分部积分法得递推公式](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/b383e1867cc14120928576bfdb115ed2~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=0SUrEgEafBdgXGrbgi6OygK6ngA%3D)
App04:45![[高数141]求极限问题|利用等价无穷小替换和常见的极限](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/3c63f99ea0e3441eb9c8fbdf1fc9070a~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=t7ADnlUNvlMS5Wgg5yqEU6JMBAE%3D)
App05:11![[高数142]求极限问题|三种方法求无穷个无穷小量的和的极限](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/572c41cb33904509866bc34f52c7d29e~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=MHm4qDWDxn892jt8%2BxYTOp0XAMY%3D)
App12:06![[高数143]求极限问题|两数列乘积的均值的极限为两数列极限的乘积](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/f84cb6d5302d449b8dd9571bad01b44c~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=bp40PLHZVjPd4MET68ri%2BGXfRwM%3D)
App08:45![[高数144]求极限|利用欧拉常数相关极限和Stirling公式](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/ecdd2e9f0f8f426fb1d02d57c4648262~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=n3oETWT%2BIQuA9JKxwipSreL5BR8%3D)
App09:18![[高数145]求极限|取整函数相关的极限问题](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/3934a7553c934b69acdb94d4fa4db1a9~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=q1xG32lRhlICbteGqEg5vxjOQ8U%3D)
App02:16![[高数146]求极限|两种方法求和式的极限问题](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/aabd864dab8946d1bab91a70f6d81834~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=sByjMUfDVpuXKWoPAf%2F5EFKFYys%3D)
App06:43![[高数147]求极限|奇数相乘/偶数相乘的极限|巧用均值不等式](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/2e3c6b47f5b44e1eabfa1de9d9522755~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=nDJVKHH5cDtT6usqVXu9AAwvSZg%3D)
App05:16![[高数148]求极限|两边夹准则求极限](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/1549ec15ebed4c11b85346d4aaa2f673~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=%2BXuefNDZH8BU6NKkPGtGJnELIjc%3D)
App04:07![[高数149]求极限|两边夹准则+已知极限](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/56b1ae4dabf7456aa5732b982835f94f~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=mk2N2wAA7nWgOd6kiUf0ZKDW944%3D)
App03:26![[高数150]求极限|积分上限函数极限问题|洛必达法则](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/9a060b86a0734da799c31304021291f6~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=Q2sQPmemmHww4%2B3UT7ibfGBFrfw%3D)
App03:38![[高数151]求极限|等价无穷小替换+洛必达法则](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/cdbce3ca6c364027a11196fbe56ddde3~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=4GQNxUS3xT7DXXuooxyDCmoauJ0%3D)
App03:46![[高数152]求极限|适当的化简可以减少洛必达法则的求导运算量](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/b9ad766927164691a886ad1d1903f452~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=ishJga1%2B7ZTGWG%2FIHHXxwB%2BEMBI%3D)
App04:22![[高数153]求极限|抽象函数的极限问题](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/510026c31de548deada355844c5c465f~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=HXdBCSze8BdUm%2Bq2ojwtG7VHOf4%3D)
App04:52![[高数154]求极限|抽象函数的极限问题|洛必达法则+微分中值定理](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/849eb0a500fc4c6380b30dd9b4e0f6c3~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=gY52S9OWcb55WQ6U0LHfbopNbWU%3D)
App07:18![[高数155]求极限|利用Taylor公式求极限](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/cd0bd55c4d87413e9d184e3079166694~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=e6vENm9vTX6O7z4VcmovwOwbzkI%3D)
App08:49![[高数156]求极限问题|利用Taylor公式求极限(2)](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/98b404e2f7f1445893bbecb05bfe681d~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=DDpfer8%2F8W3YyRr8K2oJDa9OP6U%3D)
App05:56![[高数157]求极限问题|利用Taylor公式求极限(3)](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/fb0283b451834da1b9c47bfda4440861~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=1Oh99Qclcfx%2Bt%2FDYObJkQ649xAg%3D)
App05:04![[高数158]求极限问题|利用定积分的定义(1)](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/2f6e428f68e24585a7da00dfdca6b1c7~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=hwMBM3KQqwtmaZf5Rd4HTMP2AGY%3D)
App05:27![[高数159]求极限问题|利用定积分的定义(2)](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/dc1310def2a0443fafb523f9aaf41976~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=elwVPptgLOLAvurwOEsPXZ9lPJs%3D)
App04:55![[高数160]求极限问题|利用定积分的定义+两边夹准则](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/f96cf433f4544601b0151dfae40065de~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=b8FbYuwBq8dFUeZVg%2ByHgr9ptcc%3D)
App05:55![[高数161]一道利用不定积分求函数表达式的题目](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/bd66185e4f5e400d81d79608a7a06c09~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=qiKVXux3prcrWbTICnz4iGdqSVk%3D)
App07:13![[高数162]被积函数是分段函数的不定积分问题](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/36c2dcd42eb54ed6b3947e1e3cb82a71~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=EXwEWXZ7aK1kEuYFkoAmK%2FvN0kc%3D)
App04:09![[高数163]被积函数带绝对值的不定积分问题](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/1e7bd3e72231482eb40ce7bc023a884c~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=KJgWLTXs0a0d52Z3iU%2B8Oi68J1k%3D)
App03:49![[高数164]隐函数的不定积分问题(1)](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/16d57214be694c16b18ddcb17ab232b5~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=%2F1ahf15hIpzZtyAFKoQUS5ox9zc%3D)
App04:54![[高数165]隐函数的不定积分问题(2)](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/bfeb34fd42b245c2b43bf283fabaac14~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=gchOPGTngQoc24H4Wn3gQp0DHW0%3D)
App04:31![[高数166]未知函数的不定积分](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/ddd321a2c39a446cb223b4153af9aa4a~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=hC76WPxXvs32xgIrFAsZKC%2Bqv4s%3D)
App03:09![[高数167]这种凑微分的方法也是少见](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/6420bd1330fd4bb280024435e2b370ab~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=VyBb%2FZoS4fee%2B5jEmNvGsWEDoq4%3D)
App04:48![[高数168]有道不难但复杂的题](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/94e15bf1e7884823982cd17080a8ec4b~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=xy3CYs5K%2FRo38ysRwRRzWAAAyHo%3D)
App08:08![[高数169]先倒代换后根式代换](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/76d32cdc203244d4bfe117ab240d0cf6~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=20250309231533416EB639024E3B9C7789&requestFrom=704&version_code=80402&x-expires=1773069333&x-signature=qeMyeXnZtlxWIBfsHtnwic0rhLY%3D)
App07:09打开App观看
