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$$$\sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)}$$$ 的积分
您的输入
求$$$\int \sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)}\, dx$$$。
解答
对 $$$c=\sqrt{2}$$$ 和 $$$f{\left(x \right)} = \tan{\left(x \right)} \sec{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$${\color{red}{\int{\sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)} d x}}} = {\color{red}{\sqrt{2} \int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}}$$
$$$\tan{\left(x \right)} \sec{\left(x \right)}$$$ 的积分为 $$$\int{\tan{\left(x \right)} \sec{\left(x \right)} d x} = \sec{\left(x \right)}$$$:
$$\sqrt{2} {\color{red}{\int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}} = \sqrt{2} {\color{red}{\sec{\left(x \right)}}}$$
因此，
$$\int{\sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)} d x} = \sqrt{2} \sec{\left(x \right)}$$
加上积分常数：
$$\int{\sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)} d x} = \sqrt{2} \sec{\left(x \right)}+C$$
答案
$$$\int \sqrt{2} \tan{\left(x \right)} \sec{\left(x \right)}\, dx = \sqrt{2} \sec{\left(x \right)} + C$$$A