Integral of $$$\left(\frac{1003}{1000}\right)^{x}$$$

Find $$$\int \left(\frac{1003}{1000}\right)^{x}\, dx$$$.

Apply the exponential rule $$$\int{a^{x} d x} = \frac{a^{x}}{\ln{\left(a \right)}}$$$ with $$$a=\frac{1003}{1000}$$$:

$${\color{red}{\int{\left(\frac{1003}{1000}\right)^{x} d x}}} = {\color{red}{\frac{\left(\frac{1003}{1000}\right)^{x}}{\ln{\left(\frac{1003}{1000} \right)}}}}$$

Therefore,

$$\int{\left(\frac{1003}{1000}\right)^{x} d x} = \frac{\left(\frac{1003}{1000}\right)^{x}}{\ln{\left(\frac{1003}{1000} \right)}}$$

Simplify:

$$\int{\left(\frac{1003}{1000}\right)^{x} d x} = \frac{\left(\frac{1003}{1000}\right)^{x}}{- 3 \ln{\left(10 \right)} + \ln{\left(1003 \right)}}$$

Add the constant of integration:

$$\int{\left(\frac{1003}{1000}\right)^{x} d x} = \frac{\left(\frac{1003}{1000}\right)^{x}}{- 3 \ln{\left(10 \right)} + \ln{\left(1003 \right)}}+C$$

$$$\int \left(\frac{1003}{1000}\right)^{x}\, dx = \frac{\left(\frac{1003}{1000}\right)^{x}}{- 3 \ln\left(10\right) + \ln\left(1003\right)} + C$$$A