$$$\left(x + 3\right) \ln\left(5\right)$$$ 的積分
您的輸入
求$$$\int \left(x + 3\right) \ln\left(5\right)\, dx$$$。
解答
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\ln{\left(5 \right)}$$$ 與 $$$f{\left(x \right)} = x + 3$$$：
$${\color{red}{\int{\left(x + 3\right) \ln{\left(5 \right)} d x}}} = {\color{red}{\ln{\left(5 \right)} \int{\left(x + 3\right)d x}}}$$
逐項積分：
$$\ln{\left(5 \right)} {\color{red}{\int{\left(x + 3\right)d x}}} = \ln{\left(5 \right)} {\color{red}{\left(\int{3 d x} + \int{x d x}\right)}}$$
配合 $$$c=3$$$，應用常數法則 $$$\int c\, dx = c x$$$：
$$\ln{\left(5 \right)} \left(\int{x d x} + {\color{red}{\int{3 d x}}}\right) = \ln{\left(5 \right)} \left(\int{x d x} + {\color{red}{\left(3 x\right)}}\right)$$
套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=1$$$：
$$\ln{\left(5 \right)} \left(3 x + {\color{red}{\int{x d x}}}\right)=\ln{\left(5 \right)} \left(3 x + {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}\right)=\ln{\left(5 \right)} \left(3 x + {\color{red}{\left(\frac{x^{2}}{2}\right)}}\right)$$
因此，
$$\int{\left(x + 3\right) \ln{\left(5 \right)} d x} = \left(\frac{x^{2}}{2} + 3 x\right) \ln{\left(5 \right)}$$
化簡：
$$\int{\left(x + 3\right) \ln{\left(5 \right)} d x} = \frac{x \left(x + 6\right) \ln{\left(5 \right)}}{2}$$
加上積分常數：
$$\int{\left(x + 3\right) \ln{\left(5 \right)} d x} = \frac{x \left(x + 6\right) \ln{\left(5 \right)}}{2}+C$$
答案
$$$\int \left(x + 3\right) \ln\left(5\right)\, dx = \frac{x \left(x + 6\right) \ln\left(5\right)}{2} + C$$$A