Indefinite Integration
Exponential and Trigonometric Functions
Grade 12
Question:
<p>Evaluate \(\int \frac{e^x dx}{\sin x + 1}\) (alternative form)</p>
<p>(a) \(\log e 5\)</p>
<p>(b) \(\frac{1}{4}\log\left|\frac{\tan x - 2}{\tan x + 2}\right| + C\)</p>
<p>(c) (c) \(C - \frac{e^x}{1 + \sin x}\)</p>
<p>(d) \(C - \frac{e^x \cos x}{1 + \sin x}\)</p>
Step-by-Step Solution
Key Concept: The structure of $e^x$ with trigonometric functions suggests using integration by parts after simplification.
<p>Similar to the previous integral, rationalize and use integration by parts to obtain $C - \frac{e^x \cos x}{1 + \sin x}$.</p>
Correct Answer: D