Question

Find : \[ I = \int \frac{x + \sin x}{1 + \cos x} \, dx \]

Hint:

Use trigonometric identities to simplify complex integrals involving trigonometric functions, especially when dealing with expressions like \( 1 + \cos x \).

Answer 1

First, simplify the given integral. Start by using the trigonometric identity: \[ 1 + \cos x = 2 \cos^2 \left( \frac{x}{2} \right) \] Thus, the integral becomes: \[ I = \int \frac{x + \sin x}{2 \cos^2 \left( \frac{x}{2} \right)} \, dx \] Now, split the terms in the numerator: \[ I = \frac{1}{2} \int \frac{x}{\cos^2 \left( \frac{x}{2} \right)} \, dx + \frac{1}{2} \int \frac{\sin x}{\cos^2 \left( \frac{x}{2} \right)} \, dx \] The second term can be simplified using the identity \( \sin x = 2 \sin \left( \frac{x}{2} \right) \cos \left( \frac{x}{2} \right) \). After further simplifications, integrate the expression.