$$$x \tan{\left(3 \right)}$$$の積分

$$$\int x \tan{\left(3 \right)}\, dx$$$ を求めよ。

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\tan{\left(3 \right)}$$$ と $$$f{\left(x \right)} = x$$$ に対して適用する:

$${\color{red}{\int{x \tan{\left(3 \right)} d x}}} = {\color{red}{\tan{\left(3 \right)} \int{x d x}}}$$

$$$n=1$$$ を用いて、べき乗の法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:

$$\tan{\left(3 \right)} {\color{red}{\int{x d x}}}=\tan{\left(3 \right)} {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\tan{\left(3 \right)} {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

したがって、

$$\int{x \tan{\left(3 \right)} d x} = \frac{x^{2} \tan{\left(3 \right)}}{2}$$

積分定数を加える:

$$\int{x \tan{\left(3 \right)} d x} = \frac{x^{2} \tan{\left(3 \right)}}{2}+C$$

$$$\int x \tan{\left(3 \right)}\, dx = \frac{x^{2} \tan{\left(3 \right)}}{2} + C$$$A