Question

The value of \[ \int e^{x} \left(\frac{x^{2} + 4x + 4}{(x+4)^{2}}\right)\, dx \] is:

• A. $e^{x}\left(\frac{x}{x+4}\right) + c$, where $c$ is a constant of integration.
• B. $e^{x}\left(\frac{4}{x+4}\right) + c$, where $c$ is a constant of integration.
• C. \(e^{x}\left(\frac{x}{(x+4)^{2}}\right) + C\)
• D. \(e^{x}\left(\frac{4}{(x+4)^{2}}\right) + C\)where $c$ is a constant of integration.

Hint:

When integrating expressions of the form $\int e^{x} g(x) dx$, always check if $g(x)$ can be broken down into $f(x) + f'(x)$. If it can, the integral simplifies directly to $e^{x}f(x) + c$.

Answer 1

The Correct Option is A