わざわざこんな内容を公開するのも恥ずかしいのですが、自分用の備忘録として。
三角関数
2倍角の公式
\(\sin2\alpha = 2 \sin\alpha \cos\alpha\)
\(\cos2\alpha = \cos^2\alpha-\sin^2\alpha = 2\cos^2 \alpha-1 = 1-\sin^2\alpha\)
\(\displaystyle\tan2\alpha = \frac{2\tan\alpha}{1-\tan^2\alpha} \)
3倍角の公式
\(\sin3\alpha = 3\sin\alpha-4\sin^3\alpha \)
\(\cos3\alpha = 4\cos^3\alpha-3\cos\alpha\)
半角の公式
\(\displaystyle\sin^2\frac{\alpha}{2} = \frac{1-\cos\alpha}{2}\)
\(\displaystyle\cos^2\frac{\alpha}{2} = \frac{1+\cos\alpha}{2}\)
\(\displaystyle\tan^2\frac{\alpha}{2} = \frac{1-\cos\alpha}{1+\cos\alpha}\)
微分
\(\displaystyle \frac{d}{dx} \sin x = \cos x \)
\(\displaystyle \frac{d}{dx} \cos x = \sin x \)
\(\displaystyle \frac{d}{dx} \tan x = \frac{1}{\cos^2 x} \)
積分
\(\displaystyle\int\sin x \,dx = -\cos x +C\)
\(\displaystyle\int\cos x \,dx = \sin x +C\)
\(\displaystyle\int\tan x \,dx = -\ln (|\cos{x}|) +C\)
\(\displaystyle\int\sin^2 x \,dx = \frac{x}{2} – \frac{\sin 2x}{4} +C\)
\(\displaystyle\int\cos^2 x \,dx = \frac{x}{2} + \frac{\sin 2x}{4} +C\)
\(\displaystyle\int\tan^2{x} \,dx = \tan{x}- x +C\)
\(\displaystyle\int \frac{1}{\sin x} dx = \frac{1}{2} \ln\left[ \frac{1-\cos{x}}{1+\cos{x}} \right] + C\)
\(\displaystyle\int \frac{1}{\cos x} dx = \frac{1}{2} \ln\left[ \frac{1+\sin{x}}{1-\sin{x}} \right] + C\)
\(\displaystyle\int \frac{1}{\tan x} dx = \ln(|\sin x|)+ C\)
\(\displaystyle\int \frac{1}{\sin^2 x} dx = -\frac{1}{\tan x} + C\)
\(\displaystyle\int \frac{1}{\cos^2 x} dx = \frac{1}{\tan x} + C\)
\(\displaystyle\int \frac{1}{\tan^2 x} dx = -\frac{1}{\tan x} – x+ C\)
逆三角関数
微分
\(\displaystyle \frac{d}{dx} \sin^{-1} x = \frac{1}{\sqrt{1-x^2}} \)
\(\displaystyle \frac{d}{dx} \cos^{-1} x = -\frac{1}{\sqrt{1-x^2}} \)
\(\displaystyle \frac{d}{dx} \tan^{-1} x = \frac{1}{1+x^2} \)
積分
\(\displaystyle \int\sin^{-1} x dx = x \sin^{-1} x + \sqrt{1-x^2} +C\)
\(\displaystyle \int\cos^{-1} x dx = x \cos^{-1}{x}-\sqrt{1-x^2} +C\)
\(\displaystyle \int\tan^{-1} x dx = x \tan^{-1}{x}+\frac {\ln(1+x^2)}{2} +C\)
双曲線関数
\(\displaystyle \sinh x = \frac{e^x-e^{-x}}{2}\)
\(\displaystyle \cosh x = \frac{e^x+e^{-x}}{2}\)
\(\displaystyle \tanh x = \frac{\sinh x}{\cosh x}\)
\(\displaystyle \cosh^2{x}-\sinh^2{x} = 1\)
微分
\(\displaystyle \frac{d}{dx} \sinh{x} = \cosh{x} \)
\(\displaystyle \frac{d}{dx} \cosh{x} = \sinh{x} \)
\(\displaystyle \frac{d}{dx} \tanh{x} = \frac{1}{\cosh^2{x}} \)
\(\displaystyle \frac{d^2}{dx^2} \sinh{x} = \sinh{x} \)
\(\displaystyle \frac{d^2}{dx^2} \cosh{x} = \cosh{x} \)
\(\displaystyle \frac{d^2}{dx^2} \tanh{x} = \frac{2\tanh{x}}{\cosh^2{x}} \)
積分
\(\displaystyle \int\sinh{x}\,dx = \cosh{x} +C\)
\(\displaystyle \int\cosh{x}\,dx = \sinh{x} +C\)
\(\displaystyle \int\tanh{x}\,dx = \ln{(\cosh{x})} +C\)
\(\displaystyle \int\sinh^2{x}\,dx = \frac{\cosh{2x}}{4}-\frac{x}{2} +C\)
\(\displaystyle \int\cosh^2{x}\,dx = \frac{\sinh{2x}}{4}+\frac{x}{2} +C\)
\(\displaystyle \int\tanh^2{x}\,dx = x-\tanh{x} +C\)
対数関数
微分
\(\displaystyle \frac{d}{dx} \ln x = \frac{1}{x} \)
積分
\(\displaystyle \int\ln{x}\,dx = x \ln{x}-x +C\)
\(\displaystyle \int x\ln{x}\,dx = \frac{1}{2} x^2 \ln{x} -\frac{1}{4} x^2 +C\)
\(\displaystyle \int x^2\ln{x}\,dx = \frac{1}{3} x^3 \ln{x} -\frac{1}{9} x^3 +C\)
\(\displaystyle \int (\ln{x})^2\,dx = x \left[ (\ln{x})^2 – 2 \ln{x} +2 \right] +C\)
\(\displaystyle \int \frac{\ln{x}}{x} \,dx = \frac{1}{2} (\ln{x})^2 +C\)
\(\displaystyle \int \frac{\ln{x}}{x^2} \,dx = -\frac{\ln{x}+1}{x} +C\)
指数関数
微分
\(\displaystyle \frac{d}{dx} e^x = e^x \)
積分
\(\displaystyle \int e^x \,dx = e^x +C\)
\(\displaystyle \int e^x \sin{x} \,dx = \frac{1}{2} (\sin{x}-\cos{x}) +C\)
\(\displaystyle \int e^x \cos{x} \,dx = \frac{1}{2} (\sin{x}+\cos{x}) +C\)
\(\displaystyle \int_{-\infty}^{\infty} e^{-x^2} \,dx = \sqrt{\pi} \) (\(e^{-x^2}\): ガウス関数)
\(\displaystyle \int_{-\infty}^{\infty} x^2 e^{-x^2} \,dx = \frac{\sqrt{\pi}}{2} \)
その他
\(x\)の多項式の逆数の積分
\(\displaystyle\int\frac{1}{x}dx = \ln x +C\)
\(\displaystyle\int\frac{1}{1-x^2}dx = \frac{\ln(1-x) + \ln(1+x)}{2} +C\)
\(\displaystyle\int\frac{1}{1+x^2}dx = \tan^{-1} x +C\) (\(\frac{1}{1+x^2}\): コーシー分布関数)
\(\displaystyle\int\frac{1}{1-x^3}dx = \frac{1}{6} \left[ 2\sqrt{3}\tan^{-1} \left( \frac{1+2x}{\sqrt{3}} \right) -2\ln(1-x) +\ln(1+x+x^2) \right] +C\)
\(x\)の多項式の平方逆数の積分
\(\displaystyle \int\frac{1}{\sqrt{1-x^2}}dx = \sin^{-1} x +C\)
\(\displaystyle \int\frac{1}{\sqrt{1+x^2}}dx = -\ln\left( \sqrt{1+x^2} -1 \right) +C\)
多次元釣り鐘型関数の積分
ガウス関数
\(\displaystyle \int_{-\infty}^{\infty} e^{-x^2} \,dx = \sqrt{\pi} \)
\(\displaystyle \iint_{-\infty}^{\infty} e^{-(x^2+y^2)} \,dxdy = \pi \)
\(\displaystyle \iiint_{-\infty}^{\infty} e^{-(x^2+y^2+z^2)} \,dxdydz = \pi^{\frac{3}{2}} \)
コーシー分布 (ローレンツ分布) 関数 (に類似する関数も含む)
\(\displaystyle\int_{-\infty}^{\infty} \left( 1+x^2 \right) ^{-1}dx = \pi\) (1次のコーシー分布関数)
\(\displaystyle\int_{-\infty}^{\infty} \left( 1+x^2 \right) ^{-\frac{3}{2}}dx = 2 \)
\(\displaystyle\int_{-\infty}^{\infty} \left( 1+x^2 \right) ^{-2}dx = \frac{\pi}{2}\)
\(\displaystyle\iint_{-\infty}^{\infty} \left( 1+x^2+y^2\right)^{-\frac{3}{2}}dxdy = 2\pi\) (2次のコーシー分布関数)
\(\displaystyle\iint_{-\infty}^{\infty} \left( 1+x^2+y^2\right)^{-2}dxdy = \pi\)
\(\displaystyle\iint_{-\infty}^{\infty} \left( 1+x^2+y^2\right)^{-\frac{5}{2}}dxdy = \frac{2\pi}{3}\)
\(\displaystyle\iiint_{-\infty}^{\infty} \left( 1+x^2+y^2+z^2\right)^{-2}dxdydz = \pi^2\) (3次のコーシー分布関数)
\(\displaystyle\iiint_{-\infty}^{\infty} \left( 1+x^2+y^2+z^2\right)^{-\frac{5}{2}}dxdydz = \frac{4\pi}{3}\)
\(\displaystyle\iiint_{-\infty}^{\infty} \left( 1+x^2+y^2+z^2\right)^{-3}dxdydz = \frac{\pi^2}{4}\)