tanx的n次方不定积分递推方法

递推方法

设$I_n=\int \tan^nxdx$

令$n = 0,1$,求出两个不定积分

$$\begin{array}{ll} I_0&=\int \tan^0xdx \\\\ &=\int dx \\\\ &=x+C \\\\\\ \end{array} \begin{array}{ll} I_1&=\int \tan xdx \\\\ &=\int \frac{\sin x}{\cos x}dx \\\\ &=-\int \frac{d\cos x}{\cos x} \\\\ &=-\ln |\cos x|+C \end{array}$$

递推过程

$$\begin{array}{ll} I_n&=\int \tan^nxdx \\\\ &=\int \tan^{n-2}x\cdot \tan^2xdx \\\\ &=\int \tan^{n-2}x\cdot (\sec^2x-1)dx \\\\ &=\int \tan^{n-2}x\cdot \sec^2xdx-\int \tan^{n-2}xdx \\\\ &=\int \tan^{n-2}xd\tan x-\int \tan^{n-2}xdx \\\\ &=\frac{\tan^{n-1}x}{n-1}-\int \tan^{n-2}xdx \end{array}$$