微积分公式技巧解答

考无忧论坛-----考霸整理版Abstract:Basedonthecomprehensiveanalysisontheplasticpart’sstructureservicerequirement,moundingqualityandmouldmenufactoringcost.Acorrespondinginjectionmouldofinternalsidecorepullingwasdesigned.Byadoptingthemulti-directionandmulti-combinationcore-pulling.Acorrespondinginjectionmouldofinternalsidecorepullingwasdesigned,theworkingprocessofthemouldwasintroduced有关高等数学计算过程中所涉及到的数学公式（集锦）一、00101101lim0nnnmmxmanmbaxaxanmbxbxbnm（系数不为0的情况）二、重要公式（1）0sinlim1xxx（2）10lim1xxxe（3）lim()1nnaao（4）lim1nnn（5）limarctan2xx（6）limtan2xarcx（7）limarccot0xx（8）limarccotxx（9）lim0xxe（10）limxxe（11）0lim1xxx三、下列常用等价无穷小关系（0x）sinxxtanxxarcsinxxarctanxx211cos2xxln1xx1xex1lnxaxa11xx四、导数的四则运算法则uvuvuvuvuv2uuvuvvv五、基本导数公式⑴0c⑵1xx⑶sincosxx⑷cossinxx⑸2tansecxx⑹2cotcscxx考无忧论坛-----考霸整理版⑺secsectanxxx⑻csccsccotxxx⑼xxee⑽lnxxaaa⑾1lnxx⑿1loglnxaxa⒀21arcsin1xx⒁21arccos1xx⒂21arctan1xx⒃21arccot1xx⒄1x⒅12xx六、高阶导数的运算法则（1）nnnuxvxuxvx（2）nncuxcux（3）nnnuaxbauaxb（4）()0nnnkkknkuxvxcuxvx七、基本初等函数的n阶导数公式（1）!nnxn（2）naxbnaxbeae(3)lnnxxnaaa(4)sinsin2nnaxbaaxbn(5)coscos2nnaxbaaxbn(6)11!1nnnnanaxbaxb(7)11!ln1nnnnanaxbaxb八、微分公式与微分运算法则⑴0dc⑵1dxxdx⑶sincosdxxdx⑷cossindxxdx⑸2tansecdxxdx⑹2cotcscdxxdx⑺secsectandxxxdx⑻csccsccotdxxxdx⑼xxdeedx⑽lnxxdaaadx⑾1lndxdxx⑿1loglnxaddxxa⒀21arcsin1dxdxx⒁21arccos1dxdxx考无忧论坛-----考霸整理版⒂21arctan1dxdxx⒃21arccot1dxdxx九、微分运算法则⑴duvdudv⑵dcucdu⑶duvvduudv⑷2uvduudvdvv十、基本积分公式⑴kdxkxc⑵11xxdxc⑶lndxxcx⑷lnxxaadxca⑸xxedxec⑹cossinxdxxc⑺sincosxdxxc⑻221sectancosdxxdxxcx⑼221csccotsinxdxxcx⑽21arctan1dxxcx⑾21arcsin1dxxcx十一、下列常用凑微分公式积分型换元公式1faxbdxfaxbdaxbauaxb11fxxdxfxdxux1lnlnlnfxdxfxdxxlnuxxxxxfeedxfedexue1lnxxxxfaadxfadaaxuasincossinsinfxxdxfxdxsinuxcossincoscosfxxdxfxdxcosux2tansectantanfxxdxfxdxtanux考无忧论坛-----考霸整理版2cotcsccotcotfxxdxfxdxcotux21arctanarcnarcn1fxdxftaxdtaxxarctanux21arcsinarcsinarcsin1fxdxfxdxxarcsinux十二、补充下面几个积分公式tanlncosxdxxccotlnsinxdxxcseclnsectanxdxxxccsclncsccotxdxxxc2211arctanxdxcaxaa2211ln2xadxcxaaxa221arcsinxdxcaax22221lndxxxacxa十三、分部积分法公式⑴形如naxxedx，令nux，axdvedx形如sinnxxdx令nux，sindvxdx形如cosnxxdx令nux，cosdvxdx⑵形如arctannxxdx，令arctanux，ndvxdx形如lnnxxdx，令lnux，ndvxdx⑶形如sinaxexdx，cosaxexdx令,sin,cosaxuexx均可。十四、第二换元积分法中的三角换元公式(1)22axsinxat(2)22axtanxat(3)22xasecxat【特殊角的三角函数值】（1）sin00（2）1sin62（3）3sin32（4）sin12）（5）sin0（1）cos01（2）3cos62（3）1cos32（4）cos02）（5）cos1考无忧论坛-----考霸整理版（1）tan00（2）3tan63（3）tan33（4）tan2不存在（5）tan0（1）cot0不存在（2）cot36（3）3cot33（4）cot02（5）cot不存在十五、三角函数公式1.两角和公式sin()sincoscossinABABABsin()sincoscossinABABABcos()coscossinsinABABABcos()coscossinsinABABABtantantan()1tantanABABABtantantan()1tantanABABABcotcot1cot()cotcotABABBAcotcot1cot()cotcotABABBA2.二倍角公式sin22sincosAAA2222cos2cossin12sin2cos1AAAAA22tantan21tanAAA3.半角公式1cossin22AA1coscos22AA1cossintan21cos1cosAAAAA1cossincot21cos1cosAAAAA4.和差化积公式sinsin2sincos22abababsinsin2cossin22abababcoscos2coscos22abababcoscos2sinsin22abababsintantancoscosababab考无忧论坛-----考霸整理版5.积化和差公式1sinsincoscos2ababab1coscoscoscos2ababab1sincossinsin2ababab1cossinsinsin2ababab6.万能公式22tan2sin1tan2aaa221tan2cos1tan2aaa22tan2tan1tan2aaa7.平方关系22sincos1xx22secn1xtax22csccot1xx8.倒数关系tancot1xxseccos1xxcsin1csxx9.商数关系sintancosxxxcoscotsinxxx十六、几种常见的微分方程1.可分离变量的微分方程：dyfxgydx，11220fxgydxfxgydy2.齐次微分方程：dyyfdxx3.一阶线性非齐次微分方程：dypxyQxdx解为：pxdxpxdxyeQxedxc