1. **State the problem:** We need to find the integral of the function $\left(e^x - k\right)^2$ with respect to $x$. 2. **Formula and rules:** Recall that the integral of a sum is the sum of the integrals, and the integral of $e^{ax}$ is $\frac{1}{a}e^{ax} + C$. Also, expand the square before integrating: $$\left(e^x - k\right)^2 = e^{2x} - 2ke^x + k^2$$ 3. **Rewrite the integral:** $$\int \left(e^x - k\right)^2 dx = \int \left(e^{2x} - 2ke^x + k^2\right) dx$$ 4. **Integrate term-by-term:** - Integral of $e^{2x}$ is $\frac{1}{2}e^{2x}$ - Integral of $-2ke^x$ is $-2k e^x$ - Integral of $k^2$ is $k^2 x$ 5. **Combine results:** $$\int \left(e^x - k\right)^2 dx = \frac{1}{2}e^{2x} - 2k e^x + k^2 x + C$$ where $C$ is the constant of integration. **Final answer:** $$\boxed{\int \left(e^x - k\right)^2 dx = \frac{1}{2}e^{2x} - 2k e^x + k^2 x + C}$$