**Step 1: Rewrite the integrand**
We can rewrite the integrand as follows:
\(\frac{x+5}{(x+6)^2} = \frac{(x+6) - 1}{(x+6)^2} = \frac{x+6}{(x+6)^2} - \frac{1}{(x+6)^2} = \frac{1}{x+6} - \frac{1}{(x+6)^2}\)
**Step 2: Substitute back into the integral**
Now, substitute this back into the integral:
\(\int \frac{x+5}{(x+6)^2} e^x dx = \int \left(\frac{1}{x+6} - \frac{1}{(x+6)^2}\right) e^x dx\)
**Step 3: Apply integration by parts**
Recall the integration by parts formula: \(\int [f(x) + f'(x)]e^x dx = f(x)e^x + C\)
Here, let \(f(x) = \frac{1}{x+6}\). Then, \(f'(x) = -\frac{1}{(x+6)^2}\).
So, the integral becomes:
\(\int \left(\frac{1}{x+6} - \frac{1}{(x+6)^2}\right) e^x dx = \int \left[f(x) + f'(x)\right] e^x dx = f(x)e^x + C = \frac{e^x}{x+6} + C\)
Correct Answer: \(\frac{e^{x}}{x+6}+C\)
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Pedagogical Audit
Bloom's Analysis:
This is an APPLY question because the student needs to apply integration techniques and algebraic manipulation to solve the integral.
Knowledge Dimension:PROCEDURAL
Justification:The question requires the student to follow a specific procedure or algorithm to solve the integral, involving algebraic manipulation and integration techniques.
Syllabus Audit:
In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of integration techniques as covered in the textbook.