# 降次积分法

$\int_M \mathrm{d}\omega = \oint_{\partial M} \omega$

## 例子

$I_n \, = \int \cos^n (x) \, dx\!$
$= \int \cos^ {n-1} (x) \cos (x) \, dx\!$
$= \int \cos^{n-1} (x) \, d(\sin (x)) \!$
$= \cos^{n-1} (x) \sin (x) - \int \sin (x) \, d(cos^{n-1} (x))\!$
$= \cos^{n-1} (x) \sin (x) + (n-1)\int \sin (x) \cos^{n-2} (x)\sin(x)\, dx\!$
$= \cos^{n-1} (x) \sin (x) + (n-1)\int \cos^{n-2} (x)\sin^2 (x)\, dx\!$
$= \cos^{n-1} (x) \sin (x) + (n-1)\int \cos^{n-2} (x)(1-\cos^2 (x))\, dx\!$
$= \cos^{n-1} (x) \sin (x) + (n-1)\int \cos^{n-2} (x)\, dx - (n-1)\int \cos^n (x)\, dx\!$
$= \cos^{n-1} (x) \sin (x) + (n-1) I_{n-2} - (n-1) I_n\,$
$I_n + (n-1) I_n = \cos^{n-1} (x) \sin (x) + (n-1) I_{n-2} \,$
$n I_n = \cos^{n-1} (x) \sin (x) + (n-1) I_{n-2}\,$
$I_n = \frac{1}{n}\cos^{n-1} (x) \sin (x) + \frac{n-1}{n} I_{n-2} \,$

$\int \cos^n (x) \, dx = \frac{1}{n}\cos^{n-1} (x) \sin (x) + \frac{n-1}{n} \int \cos^{n-2} (x) \, dx\!$

$n=5\,$$I_5 = \tfrac{1}{5} \cos^4 (x) \sin (x) + \tfrac{4}{5} I_3\,$
$n=3\,$$I_3 = \tfrac{1}{3} \cos^2 (x) \sin (x) + \tfrac{2}{3} I_1\,$
$\because I_1 = \int \cos (x) \, dx = \sin (x) + C_1\,$
$\therefore I_3 = \tfrac{1}{3} \cos^2 (x) \sin (x) + \tfrac{2}{3}\sin(x) + C_2\,$$C_2 = \tfrac{2}{3} C_1\,$
$I_5 = \frac{1}{5} \cos^4 (x) \sin (x) + \frac{4}{5}\left[\frac{1}{3} \cos^2 (x) \sin (x) + \frac{2}{3} \sin (x)\right] + C\,$，C为常数

## 常见降次公式

$\int \sin^n (x) \, dx = - \frac{1}{n} \sin^{n-1} (x) \cos (x) + \frac{n-1}{n} \int \sin^{n-2} (x) \, dx\!$
$\int \tan^n (x) \, dx = \frac{1}{n-1} \tan^{n-1} (x) - \int \tan^{n-2} (x) \, dx\!$
$\int (\ln (x) )^n \, dx = x (\ln (x))^n - n \int (\ln (x))^{n-1} \, dx\!$