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

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

重寫被積函數:

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

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=2 \sqrt{2}$$$ 與 $$$f{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$$：

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

令 $$$u=\cos{\left(x \right)}$$$。

則 $$$du=\left(\cos{\left(x \right)}\right)^{\prime }dx = - \sin{\left(x \right)} dx$$$ (步驟見»)，並可得 $$$\sin{\left(x \right)} dx = - du$$$。

該積分可改寫為

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

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=-1$$$ 與 $$$f{\left(u \right)} = \frac{1}{u^{2}}$$$：

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

套用冪次法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=-2$$$：

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

回顧一下 $$$u=\cos{\left(x \right)}$$$：

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

因此，

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

加上積分常數：

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

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