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$$$- \frac{8767 x^{2}}{10000} + \frac{8767 \sin{\left(x \right)}}{10000}$$$ 的積分
相關計算器: 定積分與廣義積分計算器
您的輸入
求$$$\int \left(- \frac{8767 x^{2}}{10000} + \frac{8767 \sin{\left(x \right)}}{10000}\right)\, dx$$$。
解答
逐項積分：
$${\color{red}{\int{\left(- \frac{8767 x^{2}}{10000} + \frac{8767 \sin{\left(x \right)}}{10000}\right)d x}}} = {\color{red}{\left(- \int{\frac{8767 x^{2}}{10000} d x} + \int{\frac{8767 \sin{\left(x \right)}}{10000} d x}\right)}}$$
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{8767}{10000}$$$ 與 $$$f{\left(x \right)} = x^{2}$$$：
$$\int{\frac{8767 \sin{\left(x \right)}}{10000} d x} - {\color{red}{\int{\frac{8767 x^{2}}{10000} d x}}} = \int{\frac{8767 \sin{\left(x \right)}}{10000} d x} - {\color{red}{\left(\frac{8767 \int{x^{2} d x}}{10000}\right)}}$$
套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=2$$$：
$$\int{\frac{8767 \sin{\left(x \right)}}{10000} d x} - \frac{8767 {\color{red}{\int{x^{2} d x}}}}{10000}=\int{\frac{8767 \sin{\left(x \right)}}{10000} d x} - \frac{8767 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}}{10000}=\int{\frac{8767 \sin{\left(x \right)}}{10000} d x} - \frac{8767 {\color{red}{\left(\frac{x^{3}}{3}\right)}}}{10000}$$
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{8767}{10000}$$$ 與 $$$f{\left(x \right)} = \sin{\left(x \right)}$$$：
$$- \frac{8767 x^{3}}{30000} + {\color{red}{\int{\frac{8767 \sin{\left(x \right)}}{10000} d x}}} = - \frac{8767 x^{3}}{30000} + {\color{red}{\left(\frac{8767 \int{\sin{\left(x \right)} d x}}{10000}\right)}}$$
正弦函數的積分為 $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$：
$$- \frac{8767 x^{3}}{30000} + \frac{8767 {\color{red}{\int{\sin{\left(x \right)} d x}}}}{10000} = - \frac{8767 x^{3}}{30000} + \frac{8767 {\color{red}{\left(- \cos{\left(x \right)}\right)}}}{10000}$$
因此，
$$\int{\left(- \frac{8767 x^{2}}{10000} + \frac{8767 \sin{\left(x \right)}}{10000}\right)d x} = - \frac{8767 x^{3}}{30000} - \frac{8767 \cos{\left(x \right)}}{10000}$$
化簡：
$$\int{\left(- \frac{8767 x^{2}}{10000} + \frac{8767 \sin{\left(x \right)}}{10000}\right)d x} = - \frac{8767 \left(x^{3} + 3 \cos{\left(x \right)}\right)}{30000}$$
加上積分常數：
$$\int{\left(- \frac{8767 x^{2}}{10000} + \frac{8767 \sin{\left(x \right)}}{10000}\right)d x} = - \frac{8767 \left(x^{3} + 3 \cos{\left(x \right)}\right)}{30000}+C$$
答案
$$$\int \left(- \frac{8767 x^{2}}{10000} + \frac{8767 \sin{\left(x \right)}}{10000}\right)\, dx = - \frac{8767 \left(x^{3} + 3 \cos{\left(x \right)}\right)}{30000} + C$$$A