$$$\frac{\sqrt{2} \sin{\left(2 x \right)}}{\cos^{3}{\left(x \right)}}$$$ 的积分

求$$$\int \frac{\sqrt{2} \sin{\left(2 x \right)}}{\cos^{3}{\left(x \right)}}\, dx$$$。

改写被积函数:

$${\color{red}{\int{\frac{\sqrt{2} \sin{\left(2 x \right)}}{\cos^{3}{\left(x \right)}} d x}}} = {\color{red}{\int{\frac{2 \sqrt{2} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} d x}}}$$

对 $$$c=2 \sqrt{2}$$$ 和 $$$f{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

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

设$$$u=\cos{\left(x \right)}$$$。

则$$$du=\left(\cos{\left(x \right)}\right)^{\prime }dx = - \sin{\left(x \right)} dx$$$ (步骤见»)，并有$$$\sin{\left(x \right)} dx = - du$$$。

所以，

$$2 \sqrt{2} {\color{red}{\int{\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} d x}}} = 2 \sqrt{2} {\color{red}{\int{\left(- \frac{1}{u^{2}}\right)d u}}}$$

对 $$$c=-1$$$ 和 $$$f{\left(u \right)} = \frac{1}{u^{2}}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

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

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

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

回忆一下 $$$u=\cos{\left(x \right)}$$$:

$$2 \sqrt{2} {\color{red}{u}}^{-1} = 2 \sqrt{2} {\color{red}{\cos{\left(x \right)}}}^{-1}$$

因此，

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

加上积分常数：

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

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