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 apply the identity: \[ \frac{1 + \sin x}{1 + \cos x} = \tan \left( \frac{x}{2} \right). \] Step 2: Substitute into the integral. Replacing the expression in the integral: \[ I = \int e^x \cdot \tan \left( \frac{x}{2} \right) \, dx. \] Step 3: Use substitution. Let \( u = \frac{x}{2} \). This implies \( du = \frac{1}{2} \, dx \), so \( dx = 2 \, du \). The integral transforms to: \[ I = 2 \int e^{2u} \cdot \tan(u) \, du. \] Step 4: Solve the integral. Evaluating this standard integral yields: \[ I = e^{2u} \cdot \tan(u) + C. \] Step 5: Substitute back for \( u \). Substituting \( u = \frac{x}{2} \) back into the solution: \[ I = e^x \cdot \tan \left( \frac{x}{2} \right) + C. \]