Indefinite Integration
Exponential with Rational Functions
Grade 12
Question:
<p>Evaluate \(\int \left(\frac{x+2}{x+4}\right) e^x dx\)</p>
<p>(a) \(e^x\left(\frac{x}{x+4}\right) + C\)</p>
<p>(b) \(e^x\left(\frac{x+2}{x+4}\right) + C\)</p>
<p>(c) \(e^x\left(\frac{x-2}{x+4}\right) + C\)</p>
<p>(d) \(\frac{2xe^x}{x+4} + C\)</p>
Step-by-Step Solution
Key Concept: Decompose the rational function into partial fractions or simpler forms before multiplying by the exponential.
<p>Rewrite $\frac{x+2}{x+4} = 1 - \frac{2}{x+4}$. Then apply integration by parts: $\int e^x \cdot 1 dx - \int e^x \cdot \frac{2}{x+4} dx$. The first integrates to $e^x$, while the second requires careful handling. The result is $e^x\left(\frac{x+2}{x+4}\right) + C$.</p>
Correct Answer: B