統計学にあまりでない数学

積分法

\int \sin x d x = - \cos x + C

$\begin{equation} \int \sin x dx = -\cos x + C \end{equation}$

\int \cos x d x = \sin x + C

$\begin{equation} \int \cos x dx = \sin x + C \end{equation}$

\int \tan x d x = - \log | \cos x | + C

$\begin{equation} \int \tan x dx = -\log|\cos x| + C \end{equation}$

\int \frac{1}{\cos^{2} x} d x = \tan x + C

$\begin{equation} \int \frac{1}{\cos^2 x} dx = \tan x + C \end{equation}$

\int \frac{1}{\sqrt{a^{2} - x^{2}}} d x = \sin^{- 1} \frac{x}{a} + C

$\begin{equation} \int \frac{1}{\sqrt{a^{2}-x^{2}}}dx = \sin^{-1}\frac{x}{a} + C \end{equation}$

\int \frac{1}{a^{2} + x^{2}} d x = \frac{1}{a} \tan^{- 1} \frac{x}{a} + C

$\begin{equation} \int \frac{1}{a^{2}+x^{2}}dx = \frac{1}{a}\tan^{-1}\frac{x}{a} + C \end{equation}$

\int \frac{1}{\sqrt{x^{2} + a}} d x = \log | x + \sqrt{x^{2} + a} |

$\begin{equation} \int \frac{1}{\sqrt{x^{2}+a}}dx = \log\left|\,x + \sqrt{x^{2}+a}\,\right| \end{equation}$

\sin x = \frac{2 t}{1 + t^{2}}

$\begin{equation} \sin x = \frac{2t}{1+t^{2}} \end{equation}$

\cos x = \frac{1 - t^{2}}{1 + t^{2}}

$\begin{equation} \cos x = \frac{1-t^{2}}{1+t^{2}} \end{equation}$

d x = \frac{2}{1 + t^{2}} d t

$\begin{equation} dx = \frac{2}{1+t^{2}}dt \end{equation}$

$\ds \sqrt[n]{ax + b},~\sqrt[n]{\frac{ax + b}{cx + d}}$ を含む場合，
$t = \sqrt[n]{a x + b}$
$\begin{equation} t = \sqrt[n]{ax + b} \end{equation}$

$t = \sqrt[n]{\frac{a x + b}{c x + d}}$
$\begin{equation} t = \sqrt[n]{\frac{ax + b}{cx + d}} \end{equation}$
とおく．
$f(x, \sqrt{ax^{2}+bx+c})~~(a > 0)$ の場合，次のように変数変換する．
$\sqrt{a x^{2} + b x + c} = t - \sqrt{a} x$
$\begin{equation} \sqrt{ax^{2}+bx+c}=t-\sqrt{a} x \end{equation}$
$f(x, \sqrt{ax^{2}+bx+c})~~(a < 0)$ の場合，

$ax^{2}+bx+c=0$ の解を $\alpha, \beta (\alpha < \beta)$ とおくと，
$\sqrt{a x^{2} + b x + c} = \sqrt{- a (x - α) (x - β)} = \sqrt{- a} (x - α) \sqrt{\frac{β - x}{x - α}}$
$\begin{equation} \sqrt{ax^{2}+bx+c} = \sqrt{-a(x-\alpha)(x-\beta)} =\sqrt{-a}(x-\alpha)\sqrt{\frac{\beta-x}{x-\alpha}} \end{equation}$
と変形して，次のように変数変換する．
$t = \sqrt{\frac{x - β}{x - α}}$
$\begin{equation} t = \sqrt{\frac{x - \beta}{x - \alpha}} \end{equation}$