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[高数122] 两次换元求解 | 先凑微分换元后三角代换
高等数学习题讲解 第122题 两次换元求不定积分. 1. 凑微分得 dx^4, 将 x^4=u 进行第一次换元 2. 三角代换, 将 u=tant, 进行第二次换元 最终转化为易于求解的形式.
高等数学(上)习题集104/264
![[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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=WavcPQCL5A8N7vwupUK4z4Zc79U%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=sYGd3yxrqCk6dcqnfHSc8D7ldGY%3D)
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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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=nIK20s5uqrvpweczy2RVaeUBve0%3D)
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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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=meKi7DMOjkzyy7HvGKpCXBslCe4%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://p3-sign.bdxiguaimg.com/tos-cn-i-0004/5f6adb68603b44b1b154b7770677be6c~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=kQVDFAlG3dTbU1NIrj7dux9gurw%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=AVDNDtyzyqdi0ybx2ZLkqLYJyNw%3D)
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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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=73QQOceHksbxnEnZ%2FclBAkJs3Po%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://p26-sign.bdxiguaimg.com/tos-cn-i-0004/b39d02ebb23b4465b39a0a80f8ac32d7~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=ngP%2BmVJkQOMOfOUNEwugI73zcxo%3D)
App05:23![[高数132] 少见的换元方式|换元后分部积分法求解](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/a0d0cb73da794ac0b7aded29d90e4e57~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=XtCcBDOMdmMaPrUlBPKTNi%2Fh0Rg%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=RfoE%2FbkRNYKat5deURp71lUGSL4%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=Wu2yenrKaOfa93Lxf1pQh%2BWZ5%2BU%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://p3-sign.bdxiguaimg.com/tos-cn-i-0004/b6e002f18ba948869837965c0878cf2b~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=U54P%2BjepoKI1BE0K0uNTZRFlq6s%3D)
App02:52![[高数136] 先求表达式|后求不定积分|换元法+分部积分法](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/c1688590d82748d7835fe647e74eb1d5~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=04BgFlfwD3ud8Q51btlz8F3hGq0%3D)
App06:36![[高数137] 挺简单的一道题|前提是记得cosx的任意阶导数公式](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/131053a91bc647fd98236c8fbda23f74~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=f6B%2Bt3OarJeiWe6i%2Fpi1qjWWLFI%3D)
App01:51![[高数138] 三角恒等式+分部积分法得递推公式](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/f15b85149a964e0a83eb81d9e3c8dbcc~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=mSOvqve4RDRd%2BA1NCJc7JHz2duY%3D)
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App04:55![[高数140] 三角变换+分部积分法得递推公式](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/b383e1867cc14120928576bfdb115ed2~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=8nJ42FWwhKNVbAT4811I83OYS1Y%3D)
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App09:18![[高数145]求极限|取整函数相关的极限问题](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/3934a7553c934b69acdb94d4fa4db1a9~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=ixT9foxZdgA1nvKyzpxIVG1xokU%3D)
App02:16![[高数146]求极限|两种方法求和式的极限问题](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/aabd864dab8946d1bab91a70f6d81834~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=GQF34Hw%2FQFTkT92gPdV0FDK5PaQ%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=2mnT84C4Yox1811d1%2Bnfimf5rv4%3D)
App05:16![[高数148]求极限|两边夹准则求极限](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/1549ec15ebed4c11b85346d4aaa2f673~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=FOlpe2uVV0CF1RG0hDvE9yGRbcY%3D)
App04:07![[高数149]求极限|两边夹准则+已知极限](http://p26-sign.bdxiguaimg.com/tos-cn-i-0004/56b1ae4dabf7456aa5732b982835f94f~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=6QNITvn5vbNkQ4P37L3d%2B4sT%2Bxg%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://p3-sign.bdxiguaimg.com/tos-cn-i-0004/9a060b86a0734da799c31304021291f6~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=Dc9Hb2bxqx7%2FGumOoVxGDGHyeJc%3D)
App03:38![[高数151]求极限|等价无穷小替换+洛必达法则](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/cdbce3ca6c364027a11196fbe56ddde3~tplv-pk90l89vgd-crop-center:576:324.jpeg?appId=3586&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=2024110608035221AAA7A927B344560297&requestFrom=704&version_code=80402&x-expires=1762387433&x-signature=7ICqGVLOX52vuDz%2F61AM15VBXuA%3D)
App03:46打开App观看
