$$$\frac{1}{4 \cos^{2}{\left(x \right)}}$$$の積分
入力内容
$$$\int \frac{1}{4 \cos^{2}{\left(x \right)}}\, dx$$$ を求めよ。
解答
定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\frac{1}{4}$$$ と $$$f{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$$$ に対して適用する:
$${\color{red}{\int{\frac{1}{4 \cos^{2}{\left(x \right)}} d x}}} = {\color{red}{\left(\frac{\int{\frac{1}{\cos^{2}{\left(x \right)}} d x}}{4}\right)}}$$
被積分関数を正割関数で表しなさい:
$$\frac{{\color{red}{\int{\frac{1}{\cos^{2}{\left(x \right)}} d x}}}}{4} = \frac{{\color{red}{\int{\sec^{2}{\left(x \right)} d x}}}}{4}$$
$$$\sec^{2}{\left(x \right)}$$$ の不定積分は $$$\int{\sec^{2}{\left(x \right)} d x} = \tan{\left(x \right)}$$$ です:
$$\frac{{\color{red}{\int{\sec^{2}{\left(x \right)} d x}}}}{4} = \frac{{\color{red}{\tan{\left(x \right)}}}}{4}$$
したがって、
$$\int{\frac{1}{4 \cos^{2}{\left(x \right)}} d x} = \frac{\tan{\left(x \right)}}{4}$$
積分定数を加える:
$$\int{\frac{1}{4 \cos^{2}{\left(x \right)}} d x} = \frac{\tan{\left(x \right)}}{4}+C$$
解答
$$$\int \frac{1}{4 \cos^{2}{\left(x \right)}}\, dx = \frac{\tan{\left(x \right)}}{4} + C$$$A