$$$\frac{\sin^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)}}$$$ 的积分

求$$$\int \frac{\sin^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)}}\, dx$$$。

将分子和分母同时乘以 $$$\cos^{2}{\left(x \right)}$$$，并将 $$$\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$$ 转换为 $$$\tan^{2}{\left(x \right)}$$$:

$${\color{red}{\int{\frac{\sin^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)}} d x}}} = {\color{red}{\int{\frac{\tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} d x}}}$$

将 $$$\frac{1}{\cos^{2}{\left(x \right)}}$$$ 转换为 $$$\sec^{2}{\left(x \right)}$$$:

$${\color{red}{\int{\frac{\tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} d x}}} = {\color{red}{\int{\tan^{2}{\left(x \right)} \sec^{2}{\left(x \right)} d x}}}$$

设$$$u=\tan{\left(x \right)}$$$。

则$$$du=\left(\tan{\left(x \right)}\right)^{\prime }dx = \sec^{2}{\left(x \right)} dx$$$ (步骤见»)，并有$$$\sec^{2}{\left(x \right)} dx = du$$$。

所以，

$${\color{red}{\int{\tan^{2}{\left(x \right)} \sec^{2}{\left(x \right)} d x}}} = {\color{red}{\int{u^{2} d u}}}$$

应用幂法则 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=2$$$：

$${\color{red}{\int{u^{2} d u}}}={\color{red}{\frac{u^{1 + 2}}{1 + 2}}}={\color{red}{\left(\frac{u^{3}}{3}\right)}}$$

回忆一下 $$$u=\tan{\left(x \right)}$$$:

$$\frac{{\color{red}{u}}^{3}}{3} = \frac{{\color{red}{\tan{\left(x \right)}}}^{3}}{3}$$

因此，

$$\int{\frac{\sin^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)}} d x} = \frac{\tan^{3}{\left(x \right)}}{3}$$

加上积分常数：

$$\int{\frac{\sin^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)}} d x} = \frac{\tan^{3}{\left(x \right)}}{3}+C$$

$$$\int \frac{\sin^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)}}\, dx = \frac{\tan^{3}{\left(x \right)}}{3} + C$$$A