Question

\(\int \mathrm{e}^x\left(\frac{x+5}{(x+6)^2}\right) \mathrm{d} x\) is

• A. \(\frac{\mathrm{e}^{x}}{(x+6)^{2}}+\mathrm{c}\), where c is the constant of integration.
• B. \(\frac{\mathrm{e}^{x}}{x+5}+\mathrm{c}\), where c is the constant of integration.
• C. \(\frac{\mathrm{e}^{x}}{(x+5)^{2}}+\mathrm{c}\), where c is the constant of integration.
• D. \(\frac{\mathrm{e}^{x}}{x+6}+\mathrm{c}\), where c is the constant of integration.

Hint:

Rewrite the fraction so the integrand has the form e^x [f(x) + f prime(x)].

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Use \(\int e^x\left[f(x) + f'(x)\right]dx = e^x f(x) + c\).

Step 2: Rewrite
\(\dfrac{x+5}{(x+6)^2} = \dfrac{(x+6) - 1}{(x+6)^2} = \dfrac{1}{x+6} - \dfrac{1}{(x+6)^2}\).

Step 3: Identify f
Take \(f(x) = \dfrac{1}{x+6}\), so \(f'(x) = -\dfrac{1}{(x+6)^2}\).

Step 4: Integrate
The integral is \(e^x f(x) + c = \dfrac{e^x}{x+6} + c\).

Final Answer:
The integral is e^x/(x + 6) + c. \[ \boxed{\text{(D) }\dfrac{\mathrm{e}^{x}}{x+6}+\mathrm{c}} \]