$$$x \left(20 x - 10\right) + \sqrt{3}$$$ 的積分
您的輸入
求$$$\int \left(x \left(20 x - 10\right) + \sqrt{3}\right)\, dx$$$。
解答
逐項積分：
$${\color{red}{\int{\left(x \left(20 x - 10\right) + \sqrt{3}\right)d x}}} = {\color{red}{\left(\int{\sqrt{3} d x} + \int{x \left(20 x - 10\right) d x}\right)}}$$
配合 $$$c=\sqrt{3}$$$，應用常數法則 $$$\int c\, dx = c x$$$：
$$\int{x \left(20 x - 10\right) d x} + {\color{red}{\int{\sqrt{3} d x}}} = \int{x \left(20 x - 10\right) d x} + {\color{red}{\sqrt{3} x}}$$
簡化被積函數:
$$\sqrt{3} x + {\color{red}{\int{x \left(20 x - 10\right) d x}}} = \sqrt{3} x + {\color{red}{\int{10 x \left(2 x - 1\right) d x}}}$$
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=10$$$ 與 $$$f{\left(x \right)} = x \left(2 x - 1\right)$$$：
$$\sqrt{3} x + {\color{red}{\int{10 x \left(2 x - 1\right) d x}}} = \sqrt{3} x + {\color{red}{\left(10 \int{x \left(2 x - 1\right) d x}\right)}}$$
Expand the expression:
$$\sqrt{3} x + 10 {\color{red}{\int{x \left(2 x - 1\right) d x}}} = \sqrt{3} x + 10 {\color{red}{\int{\left(2 x^{2} - x\right)d x}}}$$
逐項積分：
$$\sqrt{3} x + 10 {\color{red}{\int{\left(2 x^{2} - x\right)d x}}} = \sqrt{3} x + 10 {\color{red}{\left(- \int{x d x} + \int{2 x^{2} d x}\right)}}$$
套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=1$$$：
$$\sqrt{3} x + 10 \int{2 x^{2} d x} - 10 {\color{red}{\int{x d x}}}=\sqrt{3} x + 10 \int{2 x^{2} d x} - 10 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\sqrt{3} x + 10 \int{2 x^{2} d x} - 10 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=2$$$ 與 $$$f{\left(x \right)} = x^{2}$$$：
$$- 5 x^{2} + \sqrt{3} x + 10 {\color{red}{\int{2 x^{2} d x}}} = - 5 x^{2} + \sqrt{3} x + 10 {\color{red}{\left(2 \int{x^{2} d x}\right)}}$$
套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=2$$$：
$$- 5 x^{2} + \sqrt{3} x + 20 {\color{red}{\int{x^{2} d x}}}=- 5 x^{2} + \sqrt{3} x + 20 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- 5 x^{2} + \sqrt{3} x + 20 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$
因此，
$$\int{\left(x \left(20 x - 10\right) + \sqrt{3}\right)d x} = \frac{20 x^{3}}{3} - 5 x^{2} + \sqrt{3} x$$
化簡：
$$\int{\left(x \left(20 x - 10\right) + \sqrt{3}\right)d x} = \frac{x \left(20 x^{2} - 15 x + 3 \sqrt{3}\right)}{3}$$
加上積分常數：
$$\int{\left(x \left(20 x - 10\right) + \sqrt{3}\right)d x} = \frac{x \left(20 x^{2} - 15 x + 3 \sqrt{3}\right)}{3}+C$$
答案
$$$\int \left(x \left(20 x - 10\right) + \sqrt{3}\right)\, dx = \frac{x \left(20 x^{2} - 15 x + 3 \sqrt{3}\right)}{3} + C$$$A