储备一些微分与积分公式

因为某些特殊原因,,,我需要准备些公式。。。

常用微分公式

C ′ = 0 {C}' = 0 C=0 ( x a ) ′ = a x a − 1 {(x^a)}' = ax^{a-1} (xa)=axa1 ( sin ⁡ x ) ′ = cos ⁡ x {(\sin x)}' = \cos x (sinx)=cosx ( cos ⁡ x ) ′ = − sin ⁡ x {(\cos x)}' =- \sin x (cosx)=sinx ( t a n x ) ′ = s e c 2 x {(tanx)}' = sec^2x (tanx)=sec2x ( c o t x ) ′ = − c s c 2 x {(cotx)}' = -csc^2x (cotx)=csc2x ( s e c x ) ′ = s e c x t a n x {(secx)}' = secxtanx (secx)=secxtanx ( c s c x ) ′ = − c s c x c o t x {(cscx)}' =- cscxcotx (cscx)=cscxcotx ( a x ) ′ = a x l n a {(a^x)}' = a^xlna (ax)=axlna ( e x ) ′ = e x {(e^x)}' = e^x (ex)=ex ( log ⁡ a x ) ′ = 1 x l n a {(\log_{a}x)}' = \frac{1}{xlna} (logax)=xlna1 ln ⁡ x = 1 x \ln x = \frac{1}{x} lnx=x1 d ( u ± v ) = d u ± d v d(u\pm v) = du\pm dv d(u±v)=du±dv d ( u v ) = v d u + u d v d(uv) = vdu+udv d(uv)=vdu+udv d ( u v ) = v d u − u d v v 2 d(\frac{u}{v}) = \frac{vdu-udv}{v^2} d(vu)=v2vduudv


常用不定积分公式

∫ ( f ( x ) ± g ( x ) ) d x = ∫ f ( x ) d x ± ∫ g ( x ) d x \int (f(x) \pm g(x))dx= \int f(x)dx \pm \int g(x)dx (f(x)±g(x))dx=f(x)dx±g(x)dx ∫ k f ( x ) d x = k ∫ f ( x ) d x \int kf(x)dx=k \int f(x)dx kf(x)dx=kf(x)dx ∫ x a d x = 1 a + 1 x a + 1 + C \int x^adx=\frac{1}{a+1}x^{a+1}+C xadx=a+11xa+1+C ∫ 1 x d x = l n ∣ x ∣ + C \int \frac{1}{x}dx=ln|x|+C x1dx=lnx+C ∫ a x d x = a x l n a + C \int a^xdx=\frac{a^x}{lna}+C axdx=lnaax+C ∫ e x d x = e x + C \int e^xdx=e^x+C exdx=ex+C ∫ c o s x d x = s i n x + C \int cosxdx=sinx+C cosxdx=sinx+C ∫ s i n x d x = − c o s x + C \int sinxdx=-cosx+C sinxdx=cosx+C ∫ t a n x d x = − l n ∣ c o s x ∣ + C \int tanxdx=-ln|cosx|+C tanxdx=lncosx+C

差不多这些够用了。

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