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$$$\sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)}$$$の積分
関連する計算機: 定積分・広義積分計算機
入力内容
$$$\int \sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)}\, dx$$$ 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、$$$c=\sqrt{2}$$$ と $$$f{\left(x \right)} = \tan{\left(x \right)} \sec{\left(x \right)}$$$ に対して適用する:
$${\color{red}{\int{\sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)} d x}}} = {\color{red}{\sqrt{2} \int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}}$$
$$$\tan{\left(x \right)} \sec{\left(x \right)}$$$ の不定積分は $$$\int{\tan{\left(x \right)} \sec{\left(x \right)} d x} = \sec{\left(x \right)}$$$ です:
$$\sqrt{2} {\color{red}{\int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}} = \sqrt{2} {\color{red}{\sec{\left(x \right)}}}$$
したがって、
$$\int{\sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)} d x} = \sqrt{2} \sec{\left(x \right)}$$
積分定数を加える:
$$\int{\sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)} d x} = \sqrt{2} \sec{\left(x \right)}+C$$
解答
$$$\int \sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)}\, dx = \sqrt{2} \sec{\left(x \right)} + C$$$A