$$$\left(\frac{1003}{1000}\right)^{x}$$$ 的積分
您的輸入
求$$$\int \left(\frac{1003}{1000}\right)^{x}\, dx$$$。
解答
Apply the exponential rule $$$\int{a^{x} d x} = \frac{a^{x}}{\ln{\left(a \right)}}$$$ with $$$a=\frac{1003}{1000}$$$:
$${\color{red}{\int{\left(\frac{1003}{1000}\right)^{x} d x}}} = {\color{red}{\frac{\left(\frac{1003}{1000}\right)^{x}}{\ln{\left(\frac{1003}{1000} \right)}}}}$$
因此，
$$\int{\left(\frac{1003}{1000}\right)^{x} d x} = \frac{\left(\frac{1003}{1000}\right)^{x}}{\ln{\left(\frac{1003}{1000} \right)}}$$
化簡：
$$\int{\left(\frac{1003}{1000}\right)^{x} d x} = \frac{\left(\frac{1003}{1000}\right)^{x}}{- 3 \ln{\left(10 \right)} + \ln{\left(1003 \right)}}$$
加上積分常數：
$$\int{\left(\frac{1003}{1000}\right)^{x} d x} = \frac{\left(\frac{1003}{1000}\right)^{x}}{- 3 \ln{\left(10 \right)} + \ln{\left(1003 \right)}}+C$$
答案
$$$\int \left(\frac{1003}{1000}\right)^{x}\, dx = \frac{\left(\frac{1003}{1000}\right)^{x}}{- 3 \ln\left(10\right) + \ln\left(1003\right)} + C$$$A