高数微积分基本公式大全

高等数学微积分公式大全一、基本导数公式⑴()0c′=⑵1xxµµµ−=⑶()sincosxx′=⑷()cossinxx′=−⑸()2tansecxx′=⑹()2cotcscxx′=−⑺()secsectanxxx′=⋅⑻()csccsccotxxx′=−⋅⑼()xxee′=⑽()lnxxaaa′=⑾()1lnxx′=⑿()1loglnxaxa′=⒀()21arcsin1xx′=−⒁()21arccos1xx′=−−⒂()21arctan1xx′=+⒃()21arccot1xx′=−+⒄()1x′=⒅()12xx′=二、导数的四则运算法则()uvuv′′′±=±()uvuvuv′′′=+2uuvuvvv′′′−⎛⎞=⎜⎟⎝⎠三、微分公式与微分运算法则⑴()0dc=⑵()1dxxdxµµµ−=⑶()sincosdxxdx=⑷()cossindxxdx=−⑸()2tansecdxxdx=⑹()2cotcscdxxdx=−⑺()secsectandxxxdx=⋅⑻()csccsccotdxxxdx=−⋅⑼()xxdeedx=⑽()lnxxdaaadx=⑾()1lndxdxx=⑿()1loglnxaddxxa=⒀()21arcsin1dxdxx=−⒁()21arccos1dxdxx=−−⒂()21arctan1dxdxx=+⒃()21arccot1dxdxx=−+四、微分运算法则⑴()duvdudv±=±⑵()dcucdu=⑶()duvvduudv=+⑷2uvduudvdvv−⎛⎞=⎜⎟⎝⎠五、基本积分公式⑴kdxkxc=+∫⑵11xxdxcµµµ+=++∫⑶lndxxcx=+∫⑷lnxxaadxca=+∫⑸xxedxec=+∫⑹cossinxdxxc=+∫⑺sincosxdxxc=−+∫⑻221sectancosdxxdxxcx==+∫∫⑼221csccotsinxdxxcx==−+∫∫⑽21arctan1dxxcx=++∫⑾21arcsin1dxxcx=+−∫六、补充积分公式tanlncosxdxxc=−+∫cotlnsinxdxxc=+∫seclnsectanxdxxxc=++∫csclncsccotxdxxxc=−+∫2211arctanxdxcaxaa=++∫2211ln2xadxcxaaxa−=+−+∫221arcsinxdxcaax=+−∫22221lndxxxacxa=+±+±∫七、下列常用凑微分公式积分型换元公式()()()1faxbdxfaxbdaxba+=++∫∫uaxb=+()()()11fxxdxfxdxµµµµµ−=∫∫uxµ=()()()1lnlnlnfxdxfxdxx⋅=∫∫lnux=()()()xxxxfeedxfede⋅=∫∫xue=()()()1lnxxxxfaadxfadaa⋅=∫∫xua=()()()sincossinsinfxxdxfxdx⋅=∫∫sinux=()()()cossincoscosfxxdxfxdx⋅=−∫∫cosux=()()()2tansectantanfxxdxfxdx⋅=∫∫tanux=()()()2cotcsccotcotfxxdxfxdx⋅=∫∫cotux=()()()21arctanarcnarcn1fxdxftaxdtaxx⋅=+∫∫arctanux=()()()21arcsinarcsinarcsin1fxdxfxdxx⋅=−∫∫arcsinux=八、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。九、第二换元积分法中的三角换元公式(1)22ax−令sinxat=(2)22ax+令tanxat=(3)22xa−令secxat=十、重要公式（1）0sinlim1xxx→=（2）()10lim1xxxe→+=（3）lim()1nnaao→∞=（4）lim1nnn→∞=（5）limarctan2xxπ→∞=（6）limtan2xarcxπ→−∞=−（7）limarccot0xx→∞=（8）limarccotxxπ→−∞=（9）lim0xxe→−∞=（10）limxxe→+∞=∞（11）0lim1xxx+→=（12）00101101lim0nnnmmxmanmbaxaxanmbxbxbnm−−→∞⎧=⎪⎪+++⎪=⎨+++⎪∞⎪⎪⎩LL（系数不为0的情况）十一、下列常用等价无穷小关系（0x→）sinxxtanxxarcsinxxarctanxx211cos2xx−()ln1xx+1xex−1lnxaxa−()11xx∂+−∂十二、三角函数公式1.1.1.1.两角和公式sin()sincoscossinABABAB+=+sin()sincoscossinABABAB−=−cos()coscossinsinABABAB+=−cos()coscossinsinABABAB−=+tantantan()1tantanABABAB++=−tantantan()1tantanABABAB−−=+cotcot1cot()cotcotABABBA⋅−+=+cotcot1cot()cotcotABABBA⋅+−=−2.2.2.2.二倍角公式sin22sincosAAA=2222cos2cossin12sin2cos1AAAAA=−=−=−22tantan21tanAAA=−3.3.3.3.半角公式1cossin22AA−=1coscos22AA+=1cossintan21cos1cosAAAAA−==++1cossincot21cos1cosAAAAA+==−−4.4.4.4.和差化积公式sinsin2sincos22ababab+−+=⋅sinsin2cossin22ababab+−−=⋅coscos2coscos22ababab+−+=⋅coscos2sinsin22ababab+−−=−⋅()sintantancoscosababab++=⋅5.5.5.5.积化和差公式()()1sinsincoscos2ababab=−+−−⎡⎤⎣⎦()()1coscoscoscos2ababab=++−⎡⎤⎣⎦()()1sincossinsin2ababab=++−⎡⎤⎣⎦()()1cossinsinsin2ababab=+−−⎡⎤⎣⎦6.6.6.6.万能公式22tan2sin1tan2aaa=+221tan2cos1tan2aaa−=+22tan2tan1tan2aaa=−7.7.7.7.平方关系22sincos1xx+=22secn1xtax−=22csccot1xx−=8.8.8.8.倒数关系tancot1xx⋅=seccos1xx⋅=csin1csxx⋅=