24个基本积分：
① ∫ k d x = k x + C \int k dx = kx + C ∫kdx=kx+C
② ∫ x u d x = x u + 1 u + 1 + C \int x^u dx = \frac{x^{u+1}}{u+1} + C ∫xudx=u+1xu+1+C
③ ∫ 1 x d x = ln ∣ x ∣ + C \int\frac{1}{x}dx = \ln|x| + C ∫x1dx=ln∣x∣+C
④ ∫ 1 1 + x 2 d x = arctan x + C = − a r c c o t x + C \int \frac{1}{1+x^2}dx = \arctan x + C = -arccot x + C ∫1+x21dx=arctanx+C=−arccotx+C
⑤ ∫ 1 1 − x 2 = arcsin x + C = − arccos x + C \int \frac{1}{\sqrt{1-x^2}} = \arcsin x +C = -\arccos x + C ∫1−x21=arcsinx+C=−arccosx+C
⑥ ∫ cos x d x = sin x + C \int \cos xdx = \sin x +C ∫cosxdx=sinx+C
⑦ ∫ sin x d x = − cos x + C \int \sin xdx = -\cos x + C ∫sinxdx=−cosx+C
⑧ ∫ 1 c o s 2 x d x = ∫ sec 2 x d x = tan x + C \int \frac{1}{cos^2x}dx = \int \sec^2 xdx = \tan x + C ∫cos2x1dx=∫sec2xdx=tanx+C
⑨ ∫ 1 s i n 2 x d x = ∫ csc 2 x d x = − cot x + C \int \frac{1}{sin^2x}dx = \int \csc^2 xdx = -\cot x + C ∫sin2x1dx=∫csc2xdx=−cotx+C
⑩ ∫ sec x tan x d x = sec x + C \int \sec x\tan xdx = \sec x + C ∫secxtanxdx=secx+C
⑪ ∫ csc x cot x d x = − csc x + C \int \csc x\cot xdx = -\csc x + 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 ln a + C \int a^xdx = \frac{a^x}{\ln a} + C ∫axdx=lnaax+C
⑭ ∫ s h x d x = c h x + C \int sh xdx = chx + C ∫shxdx=chx+C
⑮ ∫ c h x d x = s h x + C \int ch xdx = shx + C ∫chxdx=shx+C
⑯ ∫ tan x d x = − ln ∣ cos x ∣ + C \int \tan xdx = -\ln|\cos x| + C ∫tanxdx=−ln∣cosx∣+C
⑰ ∫ cot x d x = ln ∣ sin x ∣ + C \int \cot xdx = \ln|\sin x| + C ∫cotxdx=ln∣sinx∣+C
⑱ ∫ sec x d x = ln ∣ sec x + tan x ∣ + C \int \sec xdx = \ln|\sec x + \tan x| + C ∫secxdx=ln∣secx+tanx∣+C
⑲ ∫ csc x d x = ln ∣ c s c x − cot x ∣ + C \int \csc xdx = \ln|csc x - \cot x| + C ∫cscxdx=ln∣cscx−cotx∣+C
⑳ ∫ 1 x 2 + a 2 d x = 1 a arctan x a + C \int \frac{1}{x^2 + a^2}dx = \frac{1}{a}\arctan \frac{x}{a} + C ∫x2+a21dx=a1arctanax+C
㉑ ∫ 1 x 2 − a 2 d x = 1 2 a ln ∣ x − a x + a ∣ + C \int \frac{1}{x^2 - a^2}dx = \frac{1}{2a}\ln|\frac{x - a}{x+a}| + C ∫x2−a21dx=2a1ln∣x+ax−a∣+C
㉒ ∫ 1 a 2 − x 2 d x = arcsin x a + C \int \frac{1}{\sqrt{a^2 - x^2}}dx = \arcsin \frac{x}{a} + C ∫a2−x21dx=arcsinax+C
㉓ ∫ 1 x 2 + a 2 d x = ln ( x + x 2 + a 2 ) + C \int \frac{1}{\sqrt{x^2 + a^2}}dx = \ln(x + \sqrt{x^2 + a^2}) + C ∫x2+a21dx=ln(x+x2+a2)+C
㉔ ∫ 1 x 2 − a 2 d x = ln ∣ x + x 2 − a 2 ∣ + C \int \frac{1}{\sqrt{x^2 - a^2}}dx = \ln|x + \sqrt{x^2 - a^2}| + C ∫x2−a21dx=ln∣x+x2−a2∣+C
两个由基本积分②推导的常用积分
① ∫ 1 x d x = 2 x + C \int \frac{1}{\sqrt{x}}dx = 2\sqrt{x} + C ∫x1dx=2x+C
② ∫ 1 x 2 d x = − 1 x + C \int \frac{1}{x^2}dx = -\frac{1}{x} + C ∫x21dx=−x1+C
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