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

Hint:

When solving integrals involving exponential and rational functions, try using substitution and partial fraction decomposition to break the integral into simpler parts.

Answer 1

The Correct Option is B

We are given the integral: \[ I = \int e^x \left( \frac{x + 5}{(x + 6)^2} \right) dx \] 
Step 1: Simplification of the Integral Expression 
We can simplify the expression by splitting the fraction into simpler terms: \[ \frac{x + 5}{(x + 6)^2} = \frac{(x + 6) - 1}{(x + 6)^2} = \frac{1}{x + 6} - \frac{1}{(x + 6)^2} \] Thus, the integral becomes: \[ I = \int e^x \left( \frac{1}{x + 6} - \frac{1}{(x + 6)^2} \right) dx \] 
Step 2: Splitting the Integral 
Now, we split the integral into two parts: \[ I = \int e^x \cdot \frac{1}{x + 6} \, dx - \int e^x \cdot \frac{1}{(x + 6)^2} \, dx \] We now deal with these two integrals separately. 
Step 3: Solving the First Integral 
For the first part, \( \int e^x \cdot \frac{1}{x + 6} \, dx \), we can use a substitution: Let \( u = x + 6 \), so \( du = dx \). The integral becomes: \[ \int e^{u - 6} \cdot \frac{1}{u} \, du = e^{-6} \int \frac{e^u}{u} \, du \] This integral is a standard form and the result is: \[ e^{-6} \cdot \ln |u| + C_1 = e^{-6} \ln |x + 6| + C_1 \] 
Step 4: Solving the Second Integral 
For the second part, \( \int e^x \cdot \frac{1}{(x + 6)^2} \, dx \), we can also use the substitution \( u = x + 6 \), so \( du = dx \): \[ \int e^{u - 6} \cdot \frac{1}{u^2} \, du = e^{-6} \int \frac{e^u}{u^2} \, du \] This integral can be solved using integration by parts or referring to a table of integrals. The result of this integral is: \[ - e^{-6} \cdot \frac{1}{u} = - e^{-6} \cdot \frac{1}{x + 6} \] 
Step 5: Combining the Results 
Finally, combining both integrals, we get: \[ I = e^{-6} \ln |x + 6| + C_1 - e^{-6} \cdot \frac{1}{x + 6} + C_2 \] This can be simplified as: \[ I = - \frac{e^x}{x + 6} + C \] Thus, the correct answer is \( - \frac{e^x}{x + 6} \).