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递推方法

In=tannxdxI_n=\int \tan^nxdx

n=0,1n = 0,1,求出两个不定积分

I0=tan0xdx=dx=x+CI1=tanxdx=sinxcosxdx=dcosxcosx=lncosx+C\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

In=tannxdx=tann2xtan2xdx=tann2x(sec2x1)dx=tann2xsec2xdxtann2xdx=tann2xdtanxtann2xdx=tann1xn1tann2xdx\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}

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