Problem Statement
Evaluate the indefinite integral:
$\int e^x \left( \frac{1 + \sin x}{1 + \cos x} \right) dx$
Verified Solution & Marking Scheme
Use Half-Angle Trigonometric Identities
$\sin x = 2\sin(x/2)\cos(x/2) \quad \text{and} \quad 1 + \cos x = 2\cos^2(x/2)$
$\frac{1 + \sin x}{1 + \cos x} = \frac{1 + 2\sin(x/2)\cos(x/2)}{2\cos^2(x/2)} = \frac{1}{2\cos^2(x/2)} + \frac{2\sin(x/2)\cos(x/2)}{2\cos^2(x/2)}$
$= \frac{1}{2}\sec^2(x/2) + \tan(x/2)$
Identify f(x) and f'(x)
Let $f(x) = \tan(x/2) \implies f'(x) = \sec^2(x/2) \cdot \frac{1}{2} = \frac{1}{2}\sec^2(x/2)$.
The integrand matches the standard form $e^x [f(x) + f'(x)]$.
Apply Theorem ∫ eˣ [f(x) + f'(x)] dx = eˣ f(x) + C
$\int e^x \left[ \tan(x/2) + \frac{1}{2}\sec^2(x/2) \right] dx = e^x \tan(x/2) + C$