微分と積分の標準形

\[ \def\arraystretch{2.5}\small \begin{array}{ccc} \dfrac{dy}{dx} & y & \vphantom{\Large \displaystyle \int} \displaystyle \int y\, dx \\ \hline & \text{\normalsize 代数関数} & \\ 1 & x & \dfrac{1}{2}x^{2} + C \\ 0 & a & ax + C \\ 1 & x ± a & \dfrac{1}{2} x^{2} ± ax + C \\ a & ax & \dfrac{1}{2} ax^{2} + C \\ 2x & x^{2} & \dfrac{1}{3} x^{3} + C \\ nx^{n-1} & x^n & \dfrac{1}{n+1} x^{n+1} + C \\ -x^{-2} & x^{-1} & \log_e x + C \\ \dfrac{du}{dx} ± \dfrac{dv}{dx} ± \dfrac{dw}{dx} & u ± v ± w & \displaystyle \int u\, dx ± \int v\, dx ± \int w\, dx \\ \dfrac{dv}{dx} + v\, \dfrac{du}{dx} & uv & {\footnotesize \text{一般形は知られていない}} \\ \dfrac{1}{v^{2}} \left(v\, \dfrac{du}{dx} - u\, \dfrac{dv}{dx}\right) & \dfrac{u}{v} & {\footnotesize \text{一般形は知られていない}} \\ \dfrac{du}{dx} & u & \displaystyle ux - \int x\, du + C \\ & \text{\normalsize 指数関数と対数関数} & \\ e^x & e^x & e^x + C \\ x^{-1} & \log_e x & x(\log_e x - 1) + C \\ 0.4343 \times x^{-1} & \log_{10} x & 0.4343x (\log_e x - 1) + C \\ \displaystyle a^x \log_e a & a^x & \dfrac{a^x}{\log_e a} + C \\ & \text{\normalsize 三角関数} & \\ \cos x & \sin x & -\cos x + C \\ -\sin x & \cos x & \sin x + C \\ \sec^{2} x & \tan x & -\log_e \cos x + C \\ & \text{\normalsize 逆三角関数} & \\ \dfrac{1}{\sqrt{(1-x^{2})}} & \arcsin x & x \cdot \arcsin x + \sqrt{1 - x^{2}} + C \\ \displaystyle -\dfrac{1}{\sqrt{(1-x^{2})}} & \arccos x & x \cdot \arccos x - \sqrt{1 - x^{2}} + C \\ \dfrac{1}{1+x^{2}} & \arctan x & \begin{array}{ll} & x \cdot \arctan x \\[-10pt] & \quad - \dfrac{1}{2} \log_e (1 + x^{2}) + C \end{array} \\ & \text{\normalsize 双曲線関数} & \\ \cosh x & \sinh x & \cosh x + C \\ \sinh x & \cosh x & \sinh x + C \\ \operatorname{sech}^{2} x & \tanh x & \log_e \cosh x + C \\ & {\footnotesize \text{その他の関数}} & \\ \displaystyle -\dfrac{1}{(x + a)^{2}} & \dfrac{1}{x + a} & \log_e (x+a) + C \\ -\dfrac{x}{(a^{2} + x^{2})^{\frac{3}{2}}} & \dfrac{1}{\sqrt{a^{2} + x^{2}}} & \log_e (x + \sqrt{a^{2} + x^{2}}) + C \\ \displaystyle \mp \dfrac{b}{(a ± bx)^{2}} & \dfrac{1}{a ± bx} & ± \dfrac{1}{b} \log_e (a ± bx) + C \\ -\dfrac{3a^{2}x}{(a^{2} + x^{2})^{\frac{5}{2}}} & \dfrac{a^{2}}{(a^{2} + x^{2})^{\frac{3}{2}}} & \dfrac{x}{\sqrt{a^{2} + x^{2}}} + C \\ \displaystyle a \cdot \cos ax & \sin ax & -\dfrac{1}{a} \cos ax + C \\ -a \cdot \sin ax & \cos ax & \dfrac{1}{a} \sin ax + C \\ \displaystyle a \cdot \sec^{2}ax & \tan ax & -\dfrac{1}{a} \log_e \cos ax + C \\ \sin 2x & \sin^{2} x & \dfrac{x}{2} - \dfrac{\sin 2x}{4} + C \\ \displaystyle -\sin 2x & \cos^{2} x & \dfrac{x}{2} + \dfrac{\sin 2x}{4} + C \\ n \cdot \sin^{n-1} x \cdot \cos x & \sin^n x & \begin{array}{l} \displaystyle \displaystyle -\frac{\cos x}{n} \sin^{n-1} x \\[-5pt] \ \ \displaystyle + \frac{n-1}{n} \int \sin^{n-2} x\, dx + C \end{array} \\ -\dfrac{\cos x}{\sin^{2} x} & \dfrac{1}{\sin x} & \log_e \tan \dfrac{x}{2} + C \\ \displaystyle -\dfrac{\sin 2x}{\sin^{4} x} & \dfrac{1}{\sin^{2} x} & -\operatorname{cotan} x + C \\ \dfrac{\sin^{2} x - \cos^{2} x}{\sin^{2} x \cdot \cos^{2} x} & \dfrac{1}{\sin x \cdot \cos x} & \log_e \tan x + C \\ \begin{array}{l} n \cdot \sin mx \cdot \cos nx \\[-15pt] \ \ + m \cdot \sin nx \cdot \cos mx\end{array} & \sin mx \cdot \sin nx & \begin{array}{l}\displaystyle \frac{1}{2} \cos(m - n)x \\[-10pt] \ \ \displaystyle - \frac{1}{2} \cos(m + n)x + C\end{array} \\ 2a\cdot\sin 2ax & \sin^{2} ax & \dfrac{x}{2} - \dfrac{\sin 2ax}{4a} + C \\ -2a\cdot\sin 2ax & \cos^{2} ax & \dfrac{x}{2} + \dfrac{\sin 2ax}{4a} + C \\ \end{array} \]