积分与求导公式大全-略排版

一、导数的四则运算法则()uvuvʹ′ʹ′ʹ′±=±()uvuvuvʹ′ʹ′ʹ′=+2uuvuvvvʹ′ʹ′ʹ′−⎛⎞=⎜⎟⎝⎠---------------------------------------------------二、基本导数公式⑴()0cʹ′=⑵1xxµµµ−=⑶()sincosxxʹ′=⑷()cossinxxʹ′=−⑸()2tansecxxʹ′=⑹()2cotcscxxʹ′=−⑺()secsectanxxxʹ′=⋅⑻()csccsccotxxxʹ′=−⋅⑼()xxeeʹ′=⑽()lnxxaaaʹ′=⑾lnx()!=1x⑿()1loglnxaxaʹ′=⒀()21arcsin1xxʹ′=−⒁()21arccos1xxʹ′=−−⒂()21arctan1xxʹ′=+⒃()21arccot1xxʹ′=−+⒄()1xʹ′=⒅()12xxʹ′=-------------------------------------------------------三、高阶导数的运算法则（1）()()()()()()()nnnuxvxuxvx±=±⎡⎤⎣⎦（2）()()()()nncuxcux=⎡⎤⎣⎦（3）()()()()nnnuaxbauaxb+=+⎡⎤⎣⎦（4）()()()()()()()0nnnkkknkuxvxcuxvx−=⋅=⎡⎤⎣⎦∑-------------------------------------------------------四、基本初等函数的n阶导数公式（1）()()!nnxn=（2）()()naxbnaxbeae++=⋅(3)()()lnnxxnaaa=(4)()()sinsin2nnaxbaaxbnπ⎛⎞+=++⋅⎡⎤⎜⎟⎣⎦⎝⎠(5)()()coscos2nnaxbaaxbnπ⎛⎞+=++⋅⎡⎤⎜⎟⎣⎦⎝⎠(6)()()()11!1nnnnanaxbaxb+⋅⎛⎞=−⎜⎟+⎝⎠+(7)()()()()()11!ln1nnnnanaxbaxb−⋅−+=−⎡⎤⎣⎦+-------------------------------------------------------五、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=−+六、微分运算法则⑴()duvdudv±=±⑵()dcucdu=⑶()duvvduudv=+⑷2uvduudvdvv−⎛⎞=⎜⎟⎝⎠七、基本积分公式⑴kdxkxc=+∫⑵11xxdxcµµµ+=++∫⑶lndxxcx=+∫⑷lnxxaadxca=+∫⑸xxedxec=+∫⑹cossinxdxxc=+∫⑺sincosxdxxc=−+∫⑻221sectancosdxxdxxcx==+∫∫⑼221csccotsinxdxxcx==−+∫∫⑽21arctan1dxxcx=++∫⑾21arcsin1dxxcx=+−∫tanlncosxdxxc=−+∫cotlnsinxdxxc=+∫seclnsectanxdxxxc=++∫csclncsccotxdxxxc=−+∫2211arctanxdxcaxaa=++∫2211ln2xadxcxaaxa−=+−+∫221arcsinxdxcaax=+−∫22221lndxxxacxa=+±+±∫-------------------------------------------------------八、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=------------------------------------------------------九、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。