$$$x \left(x - 1\right)$$$の積分
入力内容
$$$\int x \left(x - 1\right)\, dx$$$ を求めよ。
解答
Expand the expression:
$${\color{red}{\int{x \left(x - 1\right) d x}}} = {\color{red}{\int{\left(x^{2} - x\right)d x}}}$$
項別に積分せよ:
$${\color{red}{\int{\left(x^{2} - x\right)d x}}} = {\color{red}{\left(- \int{x d x} + \int{x^{2} d x}\right)}}$$
$$$n=2$$$ 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:
$$- \int{x d x} + {\color{red}{\int{x^{2} d x}}}=- \int{x d x} + {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \int{x d x} + {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$
$$$n=1$$$ 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:
$$\frac{x^{3}}{3} - {\color{red}{\int{x d x}}}=\frac{x^{3}}{3} - {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\frac{x^{3}}{3} - {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$
したがって、
$$\int{x \left(x - 1\right) d x} = \frac{x^{3}}{3} - \frac{x^{2}}{2}$$
簡単化せよ:
$$\int{x \left(x - 1\right) d x} = \frac{x^{2} \left(2 x - 3\right)}{6}$$
積分定数を加える:
$$\int{x \left(x - 1\right) d x} = \frac{x^{2} \left(2 x - 3\right)}{6}+C$$
解答
$$$\int x \left(x - 1\right)\, dx = \frac{x^{2} \left(2 x - 3\right)}{6} + C$$$A