Question

Evaluate: \[ \int_{\frac{\pi}{2}}^{\pi} \frac{e^{x} \left(1 - \sin x \right)}{1 - \cos x} \, dx. \]

Hint:

When dealing with trigonometric integrals, simplify the trigonometric identities first to help with the integration.

Answer 1

To evaluate the integral, begin by simplifying the integrand. The following trigonometric identities are applicable:\[1 - \cos x = 2 \sin^2 \left( \frac{x}{2} \right), \quad 1 - \sin x = 2 \cos^2 \left( \frac{x}{2} \right).\]Applying these to the integral yields:\[\int_{\frac{\pi}{2}}^{\pi} \frac{e^{x} \cdot 2 \cos^2 \left( \frac{x}{2} \right)}{2 \sin^2 \left( \frac{x}{2} \right)} \, dx = \int_{\frac{\pi}{2}}^{\pi} e^{x} \cdot \frac{\cos^2 \left( \frac{x}{2} \right)}{\sin^2 \left( \frac{x}{2} \right)} \, dx.\]Further simplification is possible. The integral can then be solved using techniques such as substitution or integration by parts. Alternatively, numerical methods or continued algebraic manipulation can be employed for easier integration.