Question

(a) Evaluate: \[ \int_{0}^{\pi/2} e^x \left( \frac{1 + \sin x}{1 + \cos x} \right) dx. \]

Hint:

\textbf{Part (a):} Simplify the integrand by substituting trigonometric identities like \(\sin x = 1 - \cos^2 x\) or \(\cos x = 1 - \sin^2 x\), and look for symmetry in the integral limits to reduce complexity.

Answer 1

The given integral is: \[ I = \int_{0}^{\pi/2} e^x \left( \frac{1 + \sin x}{1 + \cos x} \right) dx. \] 1. Simplify the Integrand: Rewrite \( \frac{1 + \sin x}{1 + \cos x} \) using trigonometric identities: \[ 1 + \cos x = 2\cos^2\left(\frac{x}{2}\right), \quad 1 + \sin x = 2\cos^2\left(\frac{\pi}{4} - \frac{x}{2}\right). \] Substituting: \[ \frac{1 + \sin x}{1 + \cos x} = \frac{\cos^2\left(\frac{\pi}{4} - \frac{x}{2}\right)}{\cos^2\left(\frac{x}{2}\right)}. \] The integral becomes: \[ I = \int_{0}^{\pi/2} e^x \cdot \frac{\cos^2\left(\frac{\pi}{4} - \frac{x}{2}\right)}{\cos^2\left(\frac{x}{2}\right)} dx. \] Solving this integral analytically involves further substitution or numerical methods. \bigskip