$$$24 x \sec{\left(5 \right)}$$$ 的積分
您的輸入
求$$$\int 24 x \sec{\left(5 \right)}\, dx$$$。
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 $$$c=24 \sec{\left(5 \right)}$$$ 與 $$$f{\left(x \right)} = x$$$：
$${\color{red}{\int{24 x \sec{\left(5 \right)} d x}}} = {\color{red}{\left(24 \sec{\left(5 \right)} \int{x d x}\right)}}$$
套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=1$$$：
$$24 \sec{\left(5 \right)} {\color{red}{\int{x d x}}}=24 \sec{\left(5 \right)} {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=24 \sec{\left(5 \right)} {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$
因此，
$$\int{24 x \sec{\left(5 \right)} d x} = 12 x^{2} \sec{\left(5 \right)}$$
加上積分常數：
$$\int{24 x \sec{\left(5 \right)} d x} = 12 x^{2} \sec{\left(5 \right)}+C$$
答案
$$$\int 24 x \sec{\left(5 \right)}\, dx = 12 x^{2} \sec{\left(5 \right)} + C$$$A