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积分 速查表

Symbolab 数学备忘单


积分 速查表

常用积分公式

\int x^{-1}dx=\ln(x) \int \frac{1}{x} dx=\ln(x)
\int |x|dx=\frac{x\sqrt{{x}^2}}{2} \int e^{x}dx=e^{x}
\int \sin(x)dx=-\cos(x) \int \cos(x)dx=\sin(x)
\int x^{a}dx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1


三角函数积分公式

\int \sec^2(x) dx=\tan(x) \int \csc^2(x) dx =-\cot(x)
\int \frac{1}{\sin^2(x)}dx=-\cot(x) \int \frac{1}{\cos^2(x)}dx=\tan(x)


反三角函数积分公式

\int \frac{1}{{x}^2+1}dx=\arctan(x) \int \frac{-1}{{x}^2+1}dx=\arccot(x)
\int \frac{1}{\sqrt{1-{x}^2}}dx=\arcsin(x) \int \frac{-1}{\sqrt{1-{x}^2}}dx=\arccos(x)
\int \frac{1}{|x|\sqrt{{x}^2-1}} dx = \arcsec(x) \int \frac{-1}{|x|\sqrt{{x}^2-1}} dx = \arccsc(x)
\int \frac{1}{\sqrt{{x}^2+1}} dx = \arcsinh(x) \int \frac{1}{1-{x}^2} dx = \arctanh(x)
\int \frac{1}{|x|\sqrt{{x}^2+1}} dx = -\arccsch(x)


双曲函数积分公式

\int \sech^2(x) dx = \tanh(x) \int \csch^2(x) dx = (-\coth(x))
\int \cosh(x) dx = \sinh(x) \int \sinh(x) dx = \cosh(x)
\int \csch(x) dx = \ln(\tanh(\frac{x}{2})) \int \sec(x) dx = \ln(\tan(x)+\sec(x))


特殊函数的积分公式

\int \cos(\frac{{x}^2\pi}{2})dx = \C(x) \int \frac{\sin (x)}{x}dx = \Si(x)
\int \frac{\cos (x)}{x}dx = \Ci(x) \int \frac{\sinh (x)}{x}dx = \Shi(x)
\int \frac{\cosh (x)}{x}dx = \Chi(x) \int \frac{\exp (x)}{x}dx = \Ei(x)
\int \exp{-{x}^2}dx = \frac{\sqrt{\pi}}{2}\erf(x) \int \exp{{x}^2}dx = \exp{{x}^2}\F(x)
\int \sin(\frac{{x}^2\pi}{2})dx = \S(x) \int \sin({x}^2)dx = \sqrt{\frac{\pi}{2}}\S(\sqrt{\frac{2}{\pi}}x)
\int \frac{1}{\ln(x)}dx=\li(x)


不定积分运算法则

分部积分法 \int \:uv'=uv-\int \:u'v
常数积分 \int f\left(a\right)dx=x\cdot f\left(a\right)
提出常数 \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx
加法定则 \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx
在解上添加常数
\mathrm{若}\frac{dF(x)}{dx}=f(x)\mathrm{则}\int{f(x)}dx=F(x)+C
幂法则 \int x^{a}dx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1
积分换元法 \int f\left(g\left(x\right)\right)\cdot g^'\left(x\right)dx=\int f\left(u\right)du,\:\quad u=g\left(x\right)


定积分运算法则

定积分边界
\int_{a}^{b}f(x)dx=F(b)-F(a)
=\lim_{x\to b-}(F(x))-\lim _{x\to a+}(F(x))
奇函数 \mathrm{如果}\:f\left(x\right)=-f\left(-x\right)\Rightarrow\int _{-a}^{a}f(x)dx=0
无定义点
\mathrm{如果存在}\:b,\:a<b<c,\:f(b)=\mathrm{无定义},
\int_{a}^{c}\:f(x)dx=\int_{a}^{b}\:f(x)dx+\int_{b}^{c}\:f(x)dx
相同的定义点 \int _a^a\:f\left(x\right)dx=0