積分 チートシート
共通積分
\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
三角積分
\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)
弧の三角積分
\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)
双曲線積分
\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))
特殊関数の積分
\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)
不定積分の規則
部分積分
\int \:uv'=uv-\int \:u'v
定数の積分
\int f\left(a\right)dx=x\cdot f\left(a\right)
定数を除外
\int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx
和の法則
\int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx
解答に定数を追加
\frac{dF(x)}{dx}=f(x)\mathrm{\:の場合は,}\int{f(x)}dx=F(x)+C
冪乗則
\int x^{a}dx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1
置換積分
\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)
定積分の規則
定積分極限
\int_{a}^{b}f(x)dx=F(b)-F(a)
=\lim_{x\to b-}(F(x))-\lim _{x\to a+}(F(x))
奇関数
\mathrm{If}\:f\left(x\right)=-f\left(-x\right)\Rightarrow\int _{-a}^{a}f(x)dx=0
未定義の小数点
\mathrm{存在する場合は}\:b,\:a<b<c,\:f(b)=\mathrm{未定義},
\int_{a}^{c}\:f(x)dx=\int_{a}^{b}\:f(x)dx+\int_{b}^{c}\:f(x)dx
定義済みの同じ小数点
\int _a^a\:f\left(x\right)dx=0
代数
