Evaluate ∫ e^x ( (1 + sin x)/(1 + cos x) ) , dx — JEE Mathematics

We use the standard integral identity $\int e^x [f(x) + f'(x)] \, dx = e^x f(x) + C$.
Rewrite the integrand fraction:

$$\frac{1 + \sin x}{1 + \cos x} = \frac{1 + 2\sin(x/2)\cos(x/2)}{2\cos^2(x/2)} = \frac{1}{2\cos^2(x/2)} + \frac{2\sin(x/2)\cos(x/2)}{2\cos^2(x/2)}$$

$$= \frac{1}{2}\sec^2(x/2) + \tan(x/2)$$

Let $f(x) = \tan(x/2)$. Then $f'(x) = \frac{1}{2}\sec^2(x/2)$.
The expression is exactly in the form $e^x [f(x) + f'(x)]$.
Therefore, the value of the integral is $e^x \tan(x/2) + C$.

Answer: $e^x \tan(x/2) + C$