高等数学——常用不定积分公式

一、基本积分表

  1. ∫ k d x = k x + C \int kdx=kx+C kdx=kx+C
  2. ∫ x u d x = x u + 1 u + 1 + C \int x^udx= \text{\(\frac {x^{u+1}} {u+1}\)} + C xudx=u+1xu+1+C
  3. ∫ d x x = l n ∣ x ∣ + C \int \cfrac{dx}x=ln|x|+C xdx=lnx+C
  4. ∫ d x 1 + x 2 = a r c t a n x + C \int \cfrac{dx}{1+x^2}=arctanx + C 1+x2dx=arctanx+C
  5. ∫ d x ( 1 − x 2 = a r c s i n x + C \int \cfrac{dx}{\sqrt{\mathstrut {1-x^2}}}=arcsinx+C (1x2 dx=arcsinx+C
  6. ∫ c o s x d x = s i n x + C \int cosxdx=sinx+C cosxdx=sinx+C
  7. ∫ s i n x d x = − c o s x + C \int sinxdx=-cosx+C sinxdx=cosx+C
  8. ∫ d x c o s 2 x = ∫ s e c 2 x d x = t a n x + C \int \cfrac{dx}{cos^{2}x}=\int sec^{2}xdx=tanx+C cos2xdx=sec2xdx=tanx+C
  9. ∫ d x s i n 2 x = ∫ c s c 2 x d x = c o t x + C \int \cfrac{dx}{sin^{2}x}=\int csc^{2}xdx=cotx+C sin2xdx=csc2xdx=cotx+C
  10. ∫ s e c x t a n x d x = s e c x + C \int secxtanxdx=secx+C secxtanxdx=secx+C
  11. ∫ c s c x c o t x d x = c s c x + C \int cscxcotxdx=cscx+C cscxcotxdx=cscx+C
  12. ∫ e x d x = e x + C \int e^xdx=e^x+C exdx=ex+C
  13. ∫ a x d x = a x l n a + C \int a^{x}dx=\cfrac{a^x}{lna}+C axdx=lnaax+C

二、特殊积分公式

  1. ∫ sh x   d x = ch x + C \int \text{sh}x \,\text{d}x=\text{ch}x+C shxdx=chx+C(双曲积分公式 ch 2 t − sh 2 t = 1 \text{ch}^2t-\text{sh}^2t=1 ch2tsh2t=1)
  2. ∫ ch x   d x = sh x + C \int \text{ch}x \,\text{d}x=\text{sh}x+C chxdx=shx+C(双曲积分公式 ch 2 t − sh 2 t = 1 \text{ch}^2t-\text{sh}^2t=1 ch2tsh2t=1)
  3. ∫ tan x   d x = –ln ∣ cos x ∣ + C \int \text{tan}x\,\text{d}x=\text{--}\text{ln}|\text{cos}x|+C tanxdx=lncosx+C
  4. ∫ cot x   d x = ln ∣ sin x ∣ + C \int \text{cot}x\,\text{d}x=\text{ln}|\text{sin}x|+C cotxdx=lnsinx+C
  5. ∫ sec x   d x = ln ∣ sec x + tan x ∣ + C \int \text{sec}x\,\text{d}x=\text{ln}|\text{sec}x+\text{tan}x|+C secxdx=lnsecx+tanx+C
  6. ∫ csc x   d x = ln ∣ csc x − cot x ∣ + C \int \text{csc}x\,\text{d}x=\text{ln}|\text{csc}x-\text{cot}x|+C cscxdx=lncscxcotx+C
  7. ∫ d x a 2 + x 2 = 1 a arctan x a + C \int \cfrac{dx}{a^2+x^2}=\cfrac{1}{a}\text{arctan}\cfrac{x}{a}+ C a2+x2dx=a1arctanax+C
  8. ∫ d x a 2 − x 2 = 1 2 a ln ∣ x − a x + a ∣ + C \int \cfrac{dx}{a^2-x^2}=\cfrac{1}{2a}\text{ln}|\cfrac{x-a}{x+a}|+ C a2x2dx=2a1lnx+axa+C
  9. ∫ d x ( a 2 − x 2 = arcsin x a + C \int \cfrac{dx}{\sqrt{\mathstrut {a^2-x^2}}}=\text{arcsin}\cfrac{x}{a}+C (a2x2 dx=arcsinax+C
  10. ∫ d x ( x 2 + a 2 = ln ∣ x + ( x 2 + a 2 ∣ + C \int \cfrac{dx}{\sqrt{\mathstrut {x^2+a^2}}}=\text{ln}|x+\sqrt{\mathstrut {x^2+a^2}}|+C (x2+a2 dx=lnx+(x2+a2 +C
  11. ∫ d x ( x 2 − a 2 = ln ∣ x + ( x 2 − a 2 ∣ + C \int \cfrac{dx}{\sqrt{\mathstrut {x^2-a^2}}}=\text{ln}|x+\sqrt{\mathstrut {x^2-a^2}}|+C (x2a2 dx=lnx+(x2a2 +C

三、分部积分

∫ u v ′   d x = u v − ∫ u ′ v   d x \int{uv'}\,\text{d}x=uv-\int{u'v\,\text{d}x} uvdx=uvuvdx
或者
∫ u   d v = u v − ∫ v   d u \int{u}\,\text{d}v=uv-\int{v\,\text{d}u} udv=uvvdu

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