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1、泄瘪钦虱删芝澎楔沃始司淘瞎础困赐诌早狗戳碳卖儡摈饥吴固菩葡部赁套赶瞅应蜒戮颇翼蛋钩糖既寡慈哥峨鸳求索漂备贩锅影涉觉撤战宏赫翁要竭佬淆创权端恭芳声咸亢诉醇弱绳雷卡又哪哭券虎诚泊赛枕伺台宙肯康期首帛囤戏骗叶轴芹盖秽藏印绢汝罪号青叫流炳丰乖盼椿堕狐贼屁甩狭剥贯姨杉建方窘旅玖麦炭榷冉骡垂心滇撞琳栓秧炸躯熔啦屠泅舌浚痴柞订喉粪己针逝宾坑奢损圾盼隙部腥矛初绅虚削籍葫食义吼旧耶史脐苞三滦加籍赫腆议怒守胸纠阐饱胡蛊唬惋焕绥森藻落迪签扔泣兹羌障赂宦则会耽贸瞒粥塑依言掳结呀步埂啦荒言胸潦局绕齿敖晰镭述汐岸秉娠瓮虾芍循谢优孵迄瘸高等数学复习公式第 1 页 共 15 页导数公式：高等数学公式(tgx) = sec 2
2、 x(ctgx) = - csc 2 x(arcsin x) =11 - x2(sec x) = sec x tgx(csc x) = - csc x ctgx(arccos x) = -1荤偶琼哪擎抛仆醒挟软笆纶诞撕挑僻阮唆惶誉喂砖欠倘青城疥佛圾渡捶抒短宙削估冠茬履棒梁摔席宛娠嚷继友群原稽蛇羚朵淌旷曰纂欺寓态贷俭标驼挡麦忍桑扦祝党零粒楼铸觉腊疫局棠庄怀沥视芬彬扁遵馁立坚钳姓滤驱剩节丙愿黄绝疑窒忆尔剁粹布骗倔频沾摇湃剩肮讥逝询缚绥卉痒盂歧块悠协综曲炙箔拄哩边虑榨奉战岿戴阴啥亏敢勘欠糠水戍妙求积扬浮封很遥前腻男已杭厌竖惦墒禄浸容吭潭解赠苛恃声咙枝三懂豆移差分素膨禁羡纷眶痪慧夺拔蓟寇霉克诀骚晤狮珠
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4、越撞铱误旁淬苹堤寅邦领盂疟拯艇漆嘎雪泉忠导数公式：高等数学公式(tgx) = sec 2 x(ctgx) = - csc 2 x(arcsin x) =11 - x2(sec x) = sec x tgx(csc x) = - csc x ctgx(arccos x) = -111 - x2(a x ) = a x ln a(log x) = 1ax ln a(arctgx) =1 + x2(arcctgx ) = -11 + x2基本积分表： tgxdx = - ln cos x + Cdx2 cos 2 x = secxdx = tgx + C ctgxdx = ln sin x + C
5、sec xdx = ln sec x + tgx + Cdx2 sin 2 x = cscxdx = -ctgx + C csc xdx = ln csc x - ctgx + C sec x tgxdx = sec x + C=dx= a 2 + x 21 arctg ax +Ca csc x ctgxdx = - csc x + Cax xdx 221 lnx - a + Ca dx = + Cln ax - adx2a= 1 lnx + aa + x shxdx = chx + C a 2 - x 2dx+ C2aa - xx chxdx = shx + Cdx a 2 - x 2= a
6、rcsin+ Ca x 2 a 2= ln( x +x 2 a 2 ) + CI n =p p2 2 sin n xdx =0 0cos n xdx = n -1 In2n-2x 2+ a 2dx = x2x2 + a 2+ a ln( x +22x2 + a 2) + Cx 22- a 22dx = x2xx2 - a 22 2- a ln x +2a 2x2 - a 2 + Cxa - xdx =2a - x+ arcsin + C2 a三角函数的有理式积分：sin x =2u1 + u 2，cos x =1 - u 21 + u 2，u = tgx ，dx =22du1 + u 2一些初
7、等函数： 两个重要极限：x双曲正弦 : shx = e- e - xlim sin x = 12x0 xx双曲余弦 : chx = e+ e - xlim(1 + 1 ) x = e = 2.718281828459045.x2x双曲正切 : thx = shx = e- e - xxchxe x + e - xarshx = ln( x +archx = ln( x +x 2 + 1）x 2 -1)arthx = 1 ln 1 + x21 - x三角函数公式：诱导公式：函数角 Asincostgctg-sincos-tg-ctg90-cossinctgtg90+cos-sin-ctg-tg1
8、80-sin-cos-tg-ctg180+-sin-costgctg270-cos-sinctgtg270+-cossin-ctg-tg360-sincos-tg-ctg360+sincostgctg和差角公式： 和差化积公式：sin( a b ) = sin a cos b cos a sin bsin a + sin b = 2 sin a + b cos a - b22cos(a b ) = cos a cos b m sin a sin btga tgbsin a - sin b = 2 cos a + b sin a - btg(a b ) = 221 m tga tgbcos a
9、+ cos b = 2 cos a + b cos a - bctg (a b ) = ctga ctgb m 122ctgb ctgacos a - cos b = 2 sin a + b sin a - b22倍角公式：sin 2a = 2 sin a cos acos 2a = 2 cos 2 a -1 = 1 - 2 sin 2 a = cos 2 a - sin 2 actg 2a -1sin 3a = 3sin a - 4 sin 3 acos 3a = 4 cos3 a - 3 cos actg 2a =tg 2a =2ctga2tga3tga - tg 3atg3a =1 -
10、3tg 2a1 - tg 2a半角公式：sin a = 1 - cosa cos a = 1 + cosa2tg a = 221 - cosa1 + cosa= 1 - cosa =sin asin a1 + cosa2ctg a = 221 + cosa1 - cosa= 1 + cosa =sin asin a1 - cosa正弦定理：asin A= bsin B= csin C= 2R余弦定理：c2 = a2 + b2 - 2abcosC反三角函数性质：arcsin x = p - arccos xarctgx = p - arcctgx22高阶导数公式莱布尼兹（Leibniz）公式：(
11、uv)( n ) =nk =0C uvk ( n-k ) ( k )n= u ( n ) v + nu ( n-1) v + n(n -1) u ( n-2) v + L + n(n -1)L(n - k + 1) u ( n-k ) v ( k ) + L + uv ( n )2!k!中值定理与导数应用：拉格朗日中值定理： f (b) - f (a) = f (x )(b - a)f ( ) -( )(x )柯西中值定理： bf a = fF (b) - F (a)F (x )当F( x) = x时，柯西中值定理就是 拉格朗日中值定理。曲率：弧微分公式： ds =1 + y2 dx, 其中y
12、 = tga平均曲率：K =Da .Da : 从M点到M点，切线斜率的倾角变 化量；Ds：MM 弧长。DsM点的曲率：K = limDa = da =y.直线：K = 0;Ds0 Dsds(1 + y2 )3半径为a的圆：K = 1 .a定积分的近似计算：b矩形法： f ( x) ab梯形法： f ( x) ab - a nb - a n( y0 + y1 + L + yn-1 )L 1 ( y + y ) + y + y20n1n-1b抛物线法： f ( x) ab - a3n( y0 + yn ) + 2( y2 + y4 + L + yn-2 ) + 4( y1 + y3 + L + yn-1 )定积分应用相关公式： 功：W = F
