Question

The integral $ \int e^x \left( \frac{x + 5}{(x + 6)^2} \right) dx $ is:

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

Hint:

When dealing with complex integrals, look for substitution opportunities or integration by parts to simplify the expression.

Answer 1

The Correct Option is B

We are given the integral: \[ \int e^x \left( \frac{x + 5}{(x + 6)^2} \right) dx \] To solve this, we can use substitution. Let: \[ u = x + 6 \quad \Rightarrow \quad du = dx \] Thus, the integral becomes: \[ \int e^{u - 6} \left( \frac{u - 1}{u^2} \right) du \] Simplifying, we get: \[ e^{-6} \int e^u \left( \frac{u - 1}{u^2} \right) du \] By applying the method of integration by parts and solving, we end up with the result: \[ - \frac{e^x}{x + 6} \] 
Thus, the correct answer is \( - \frac{e^x}{x + 6} \).