$$$- \tan{\left(1 \right)} \tan{\left(x \right)} \sec{\left(x \right)}$$$の積分
関連する計算機: 定積分・広義積分計算機
入力内容
$$$\int \left(- \tan{\left(1 \right)} \tan{\left(x \right)} \sec{\left(x \right)}\right)\, dx$$$ 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、$$$c=- \tan{\left(1 \right)}$$$ と $$$f{\left(x \right)} = \tan{\left(x \right)} \sec{\left(x \right)}$$$ に対して適用する:
$${\color{red}{\int{\left(- \tan{\left(1 \right)} \tan{\left(x \right)} \sec{\left(x \right)}\right)d x}}} = {\color{red}{\left(- \tan{\left(1 \right)} \int{\tan{\left(x \right)} \sec{\left(x \right)} d x}\right)}}$$
$$$\tan{\left(x \right)} \sec{\left(x \right)}$$$ の不定積分は $$$\int{\tan{\left(x \right)} \sec{\left(x \right)} d x} = \sec{\left(x \right)}$$$ です:
$$- \tan{\left(1 \right)} {\color{red}{\int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}} = - \tan{\left(1 \right)} {\color{red}{\sec{\left(x \right)}}}$$
したがって、
$$\int{\left(- \tan{\left(1 \right)} \tan{\left(x \right)} \sec{\left(x \right)}\right)d x} = - \tan{\left(1 \right)} \sec{\left(x \right)}$$
積分定数を加える:
$$\int{\left(- \tan{\left(1 \right)} \tan{\left(x \right)} \sec{\left(x \right)}\right)d x} = - \tan{\left(1 \right)} \sec{\left(x \right)}+C$$
解答
$$$\int \left(- \tan{\left(1 \right)} \tan{\left(x \right)} \sec{\left(x \right)}\right)\, dx = - \tan{\left(1 \right)} \sec{\left(x \right)} + C$$$A