Indefinite Integration
Exponential and Trigonometric Functions
Grade 12
Question:
<p>Evaluate \(\int \frac{e^x dx}{\sin x + 1}\)</p>
<p>(a) \(\frac{e^x \cos x}{1 + \sin x} + C\)</p>
<p>(b) \(C - \frac{e^x \sin x}{1 + \sin x}\)</p>
<p>(c) \(C - \frac{e^x \cos x}{1 + \sin x}\)</p>
<p>(d) \(C - \frac{2e^x}{1 + \sin x}\)</p>
Step-by-Step Solution
Key Concept: Use rationalization to convert the denominator into a manageable form, then apply integration by parts.
<p>Rationalize the denominator by multiplying by $\frac{1 - \sin x}{1 - \sin x}$. This converts the integral into a form where integration by parts with $e^x$ can be applied systematically, yielding $C - \frac{e^x \cos x}{1 + \sin x}$.</p>
Correct Answer: C