Question

\[ \int \frac{x + 5}{(x + 6)^2} e^x \, dx \] is equal to:

• A. $\log(x + 6) + C$
• B. $e^x + C$
• C. $\frac{e^x}{x + 6} + C$
• D. $-\frac{1}{(x + 6)^2} e^x + C$

Hint:

When dealing with rational functions and exponential terms, try using substitution to simplify the integrand.

Answer 1

The Correct Option is C

We are given the integral: \[ \int \frac{x + 5}{(x + 6)^2} e^x \, dx. \] We can simplify this by performing substitution. Let $u = x + 6$, so that $du = dx$ and $x = u - 6$. The integral becomes: \[ \int \frac{(u - 6) + 5}{u^2} e^{u - 6} \, du. \] Simplifying further: \[ \int \frac{u - 1}{u^2} e^{u - 6} \, du. \] This can be split into two parts: \[ \int \frac{1}{u} e^{u - 6} \, du - \int \frac{1}{u^2} e^{u - 6} \, du. \] The first part gives $\frac{e^{u-6}}{u}$, and the second part involves integration by parts or recognizing standard forms, resulting in: \[ \frac{e^x}{x + 6} + C. \]