$$$\frac{1}{4 \cos^{2}{\left(x \right)}}$$$ 的積分
您的輸入
求$$$\int \frac{1}{4 \cos^{2}{\left(x \right)}}\, dx$$$。
解答
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{1}{4}$$$ 與 $$$f{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$$$：
$${\color{red}{\int{\frac{1}{4 \cos^{2}{\left(x \right)}} d x}}} = {\color{red}{\left(\frac{\int{\frac{1}{\cos^{2}{\left(x \right)}} d x}}{4}\right)}}$$
將被積函數以正割表示:
$$\frac{{\color{red}{\int{\frac{1}{\cos^{2}{\left(x \right)}} d x}}}}{4} = \frac{{\color{red}{\int{\sec^{2}{\left(x \right)} d x}}}}{4}$$
$$$\sec^{2}{\left(x \right)}$$$ 的積分是 $$$\int{\sec^{2}{\left(x \right)} d x} = \tan{\left(x \right)}$$$：
$$\frac{{\color{red}{\int{\sec^{2}{\left(x \right)} d x}}}}{4} = \frac{{\color{red}{\tan{\left(x \right)}}}}{4}$$
因此，
$$\int{\frac{1}{4 \cos^{2}{\left(x \right)}} d x} = \frac{\tan{\left(x \right)}}{4}$$
加上積分常數：
$$\int{\frac{1}{4 \cos^{2}{\left(x \right)}} d x} = \frac{\tan{\left(x \right)}}{4}+C$$
答案
$$$\int \frac{1}{4 \cos^{2}{\left(x \right)}}\, dx = \frac{\tan{\left(x \right)}}{4} + C$$$A