不定積分の基本的な計算

$$\frac{d}{dx}x=1 \qquad \Leftrightarrow\qquad \int dx=x+C$$ (680)

$$\frac{d}{dx}x^{n+1}=(n+1)x^{n} \qquad \Leftrightarrow\qquad \int x^{n}\,dx=\frac{x^{n+1}}{n+1}+C \quad(n\neq-1)$$ (681)

$$\frac{d}{dx}x^{\alpha+1}=(\alpha+1)x^{\alpha} \qquad \Leftrightarrow\qquad \int x^{\alpha}\,dx=\frac{x^{\alpha+1}}{\alpha+1}+C \quad(\alpha\in\mathbb{R},\alpha\neq-1)$$ (682)

$$\left\{\begin{array}{cc} \displaystyle{\frac{d}{dx}\log x=\frac{1... ...isplaystyle{\frac{d}{dx}\log(-x)=\frac{1}{x}} & (x<0) \end{array}\right. \qquad \Leftrightarrow\qquad \int\frac{1}{x}\,dx=\log\vert x\vert+C$$ (683)

$$\frac{d}{dx}e^{x}=e^{x} \qquad \Leftrightarrow\qquad \int e^{x}\,dx=e^{x}+C$$ (684)

$$\frac{d}{dx}a^{x}=(\log a)a^{x} \qquad \Leftrightarrow\qquad \int a^{x}\,dx=\frac{a^{x}}{\log a}+C \quad(a>0,a\neq1)$$ (685)

$$\frac{d}{dx}\cos x=-\sin x \qquad \Leftrightarrow\qquad \int\sin x\,dx=-\cos x+C$$ (686)

$$\frac{d}{dx}\sin x=\cos x \qquad \Leftrightarrow\qquad \int\cos x\,dx=\sin x+C$$ (687)

$$\frac{d}{dx}\tan x=\frac{1}{\cos^2x} \qquad \Leftrightarrow\qquad \int\frac{1}{\cos^2 x}\,dx=\tan x+C$$ (688)

$$\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^2}} \qquad \Leftrightarrow\qquad \int\frac{1}{\sqrt{1-x^2}}\,dx=\arcsin x+C \quad(\vert x\vert<1)$$ (689)

$$\frac{d}{dx}\arctan x=\frac{1}{1+x^2} \qquad \Leftrightarrow\qquad \int\frac{1}{1+x^2}\,dx=\arctan x+C$$ (690)

$$\frac{d}{dx}\cosh x=\sinh x \qquad \Leftrightarrow\qquad \int\sinh x\,dx=\cosh x+C$$ (691)

$$\frac{d}{dx}\sinh x=\cosh x \qquad \Leftrightarrow\qquad \int\cosh x\,dx=\sinh x+C$$ (692)

$$\frac{d}{dx}\tanh x=\frac{1}{\cosh^2x} \qquad \Leftrightarrow\qquad \int\frac{1}{\cosh^2x}\,dx=\tanh x+C$$ (693)

$$\frac{d}{dx}\mathrm{arcsinh}\,x=\frac{1}{\sqrt{x^2+1}} \qquad \Leftrightarrow\qquad \int\frac{1}{\sqrt{x^2+1}}\,dx=\mathrm{arcsinh}\,x+C$$ (694)

$$\frac{d}{dx}\mathrm{arccosh}\,x=\frac{1}{\sqrt{x^2-1}} \qquad \Leftrightarrow\qquad \int\frac{1}{\sqrt{x^2-1}}\,dx=\mathrm{arccosh}\,x+C \quad(\vert x\vert>1)$$ (695)

$$\frac{d}{dx}\mathrm{arctanh}\,x=\frac{1}{1-x^2} \qquad \Leftrightarrow\qquad \int\frac{1}{1-x^2}\,dx=\mathrm{arctanh}\,x+C \quad(x\neq1)$$ (696)

例 5.4  

$$\int x^8\,dx=$$ (697)

$$\int x^{\frac{1}{4}}\,dx=$$ (698)

$$\int\frac{dx}{x^3}=$$ (699)

$$\int(x^3-x^2+3x-2)\,dx=$$ (700)

Kondo Koichi
Created at 2002/09/12