The value of the integral $\int e^{x+1}(2x^2 - \frac{1}{x} + 1) dx$ is equal to (where $C$ is the constant of integration)
Step-by-Step Solution
Key Concept: Apply integration by parts to handle the product of $e^{(x+1)}$ and a rational function.
$I = \int \frac{e^{x+1}}{1} dx + \int e^{(x+1)}(2x - \frac{1}{x}) dx$. Using integration by parts on the second term with $u = 2x - \frac{1}{x}$ and $dv = e^{(x+1)} dx$, we get $I = xe^{x+1} + C$ after careful calculation of the integration by parts.
Correct Answer: C