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[高数124] 典型的三角代换思路|换元后再利用分部积分求解
高等数学习题讲解 第124题 三角代换后, 利用分部积分求解.
高等数学(上)习题集106/264
![[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://p26-sign.bdxiguaimg.com/tos-cn-i-0004/52aa9d26e9a54f388cb1b70df97997fb~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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=UEQxXWLtLaJHl85qWv5EoeBfFU4%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=tX%2F%2Bld2P41sV12pKBYh8IT3ypdk%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=fpxBsM%2BMm%2BOSd5fsxr7EUPJk%2BnM%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=G0nfDja5D9vKbRW3oq8a5G0IqpM%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=uSWeFLvPvJGOAyi9%2B30Z7DrUvms%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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=idRCVZtl7jq0U%2Bp%2FSy4rw3TGtJ8%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=Ix%2F35THONna%2B50gOaO03Fxk4DPE%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://p3-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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=5h5EUc1Vl3V99ZxjxKfjx51%2FheE%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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=KE6cd9wS1VfXj4xzWsDF7blQCsc%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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=P08pZShSXxv1WUANZpMuxdyGY3g%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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=aaYWtoNzet9fpqm%2B%2BG7bbDiAisE%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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=OEELoSsMA2fi2zAYueejLcg1%2BeY%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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=UEN6RIGkRmOC3k5xhFS6r0yJ9uM%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://p26-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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=iTQK22VZBjl1myig4ViNiZZZv9U%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://p3-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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=HfpxFiKeYKtCBD8lOvGds%2BjmC3w%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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=ZOVvsyKMFaM6Jv5edoYVWYKH2lA%3D)
App02:29![[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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=cAckeQgZax55bVToGjQ1eELzAqk%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=naXJjyfTXA0P1RAMA%2FHzUSkBo4g%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://p3-sign.bdxiguaimg.com/tos-cn-i-0004/3c63f99ea0e3441eb9c8fbdf1fc9070a~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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=zhf1RsTXUrmJGGTpnRAwTuUXQPE%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=YEZQBphMRANW%2FAaB7C6NsO%2FmtzA%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://p3-sign.bdxiguaimg.com/tos-cn-i-0004/f84cb6d5302d449b8dd9571bad01b44c~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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=augHcGH9%2FHcbqhP5U9GNzhgA21U%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=QrxP6Q0i6Bdy7BHgdMW5N0Pza4I%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=3KeAPdsEjSDeCael%2FRF9CaLXAYA%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&channelId=0&customType=custom%2Fnone&from=704_middle_image&imageType=list&isImmersiveScene=0&is_ios=1&is_stream=0&lk3s=9d3f5bff&logId=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=vM2p4tSrgO8%2BoMFDH8%2BWQfRwAZw%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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=682KDABbbWN%2FzMfMX26BQCU4ukY%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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=kh%2Bp%2BwRx%2F7%2BleqLt2rEyz%2BH2YTY%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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=oK19JyvaJbDKaUM8%2BrboimdSx8E%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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=657B7jX7N16gOFfeudSM63kkBwY%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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=0q8ABPOBWhjotucg3gTiAGzEPGw%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://p26-sign.bdxiguaimg.com/tos-cn-i-0004/b9ad766927164691a886ad1d1903f452~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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=KmlJoGi30pBrwvV%2BmpRrOEtgsoE%3D)
App04:22![[高数153]求极限|抽象函数的极限问题](http://p3-sign.bdxiguaimg.com/tos-cn-i-0004/510026c31de548deada355844c5c465f~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=20241106075557F641F04733817060E160&requestFrom=704&version_code=80402&x-expires=1762386958&x-signature=0xrVdhnaLjyqSiY9M6kpsm4AU8U%3D)
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