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

The integral \( \int e^x \left( \frac{x + 5}{(x + 6)^2} \right) dx \) can be solved using substitution. Let \( u = x + 6 \), which implies \( du = dx \). The integral transforms to \( \int e^{u - 6} \left( \frac{u - 1}{u^2} \right) du \), which simplifies to \( e^{-6} \int e^u \left( \frac{u - 1}{u^2} \right) du \). After applying integration by parts and solving, the result obtained is \( - \frac{e^x}{x + 6} \). Therefore, the correct answer is \( - \frac{e^x}{x + 6} \).