$$$5 x^{38} \left(6 x^{3} - 9\right)$$$ 的积分
您的输入
求$$$\int 5 x^{38} \left(6 x^{3} - 9\right)\, dx$$$。
解答
输入已重写为：$$$\int{5 x^{38} \left(6 x^{3} - 9\right) d x}=\int{x^{38} \left(30 x^{3} - 45\right) d x}$$$。
化简被积函数:
$${\color{red}{\int{x^{38} \left(30 x^{3} - 45\right) d x}}} = {\color{red}{\int{15 x^{38} \left(2 x^{3} - 3\right) d x}}}$$
对 $$$c=15$$$ 和 $$$f{\left(x \right)} = x^{38} \left(2 x^{3} - 3\right)$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$${\color{red}{\int{15 x^{38} \left(2 x^{3} - 3\right) d x}}} = {\color{red}{\left(15 \int{x^{38} \left(2 x^{3} - 3\right) d x}\right)}}$$
Expand the expression:
$$15 {\color{red}{\int{x^{38} \left(2 x^{3} - 3\right) d x}}} = 15 {\color{red}{\int{\left(2 x^{41} - 3 x^{38}\right)d x}}}$$
逐项积分：
$$15 {\color{red}{\int{\left(2 x^{41} - 3 x^{38}\right)d x}}} = 15 {\color{red}{\left(- \int{3 x^{38} d x} + \int{2 x^{41} d x}\right)}}$$
对 $$$c=3$$$ 和 $$$f{\left(x \right)} = x^{38}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$$15 \int{2 x^{41} d x} - 15 {\color{red}{\int{3 x^{38} d x}}} = 15 \int{2 x^{41} d x} - 15 {\color{red}{\left(3 \int{x^{38} d x}\right)}}$$
应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=38$$$：
$$15 \int{2 x^{41} d x} - 45 {\color{red}{\int{x^{38} d x}}}=15 \int{2 x^{41} d x} - 45 {\color{red}{\frac{x^{1 + 38}}{1 + 38}}}=15 \int{2 x^{41} d x} - 45 {\color{red}{\left(\frac{x^{39}}{39}\right)}}$$
对 $$$c=2$$$ 和 $$$f{\left(x \right)} = x^{41}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$$- \frac{15 x^{39}}{13} + 15 {\color{red}{\int{2 x^{41} d x}}} = - \frac{15 x^{39}}{13} + 15 {\color{red}{\left(2 \int{x^{41} d x}\right)}}$$
应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=41$$$：
$$- \frac{15 x^{39}}{13} + 30 {\color{red}{\int{x^{41} d x}}}=- \frac{15 x^{39}}{13} + 30 {\color{red}{\frac{x^{1 + 41}}{1 + 41}}}=- \frac{15 x^{39}}{13} + 30 {\color{red}{\left(\frac{x^{42}}{42}\right)}}$$
因此，
$$\int{x^{38} \left(30 x^{3} - 45\right) d x} = \frac{5 x^{42}}{7} - \frac{15 x^{39}}{13}$$
化简：
$$\int{x^{38} \left(30 x^{3} - 45\right) d x} = \frac{5 x^{39} \left(13 x^{3} - 21\right)}{91}$$
加上积分常数：
$$\int{x^{38} \left(30 x^{3} - 45\right) d x} = \frac{5 x^{39} \left(13 x^{3} - 21\right)}{91}+C$$
答案
$$$\int 5 x^{38} \left(6 x^{3} - 9\right)\, dx = \frac{5 x^{39} \left(13 x^{3} - 21\right)}{91} + C$$$A