Question

Integrate the function \( \int e^x \left( \frac{1 + \sin x}{1 + \cos x} \right) dx \):

Hint:

When simplifying integrals with trigonometric functions and exponentials, use standard trigonometric identities and substitution to simplify the integrals.

Answer 1

Step 1: Simplify the trigonometric expression. We use the identity: \[ \frac{1 + \sin x}{1 + \cos x} = \tan \left( \frac{x}{2} \right). \] Step 2: Substitute into the integral. Substituting this into the integral: \[ I = \int e^x \cdot \tan \left( \frac{x}{2} \right) \, dx. \] Step 3: Use substitution. Let \( u = \frac{x}{2} \), so \( du = \frac{1}{2} \, dx \) and \( dx = 2 \, du \). The integral becomes: \[ I = 2 \int e^{2u} \cdot \tan(u) \, du. \] Step 4: Solve the integral. This is a standard integral, and we get: \[ I = e^{2u} \cdot \tan(u) + C. \] Step 5: Substitute back for \( u \). Substitute \( u = \frac{x}{2} \) into the result: \[ I = e^x \cdot \tan \left( \frac{x}{2} \right) + C. \]