Question:

Step 1: Simplify the integrand using a trigonometric identity. We use the identity $1 + \cos x = 2\cos^2(x/2)$ to rewrite the denominator of the integrand. $$ \int \frac{dx}{1+\cos x} = \int \frac{dx}{2\cos^2(x/2)} $$ Then, we can express $\frac{1}{\cos^2(x/2)}$ as $\sec^2(x/2)$ and factor out the constant $\frac{1}{2}$. $$ \int \frac{dx}{2\cos^2(x/2)} = \frac{1}{2}\int \sec^2(x/2) \, dx $$ Step 2: Perform indefinite integration. We integrate $\frac{1}{2}\int \sec^2(x/2) \, dx$. Let $u = x/2$, so $du = \frac{1}{2} dx$. The integral becomes: $$ \frac{1}{2}\int \sec^2(x/2) \, dx = \int \sec^2(u) \, du = \tan(u) + C = \tan(x/2) + C $$ Step 3: Apply the limits of integration. Now, we apply the definite integral limits from $x = \pi/4$ to $x = 3\pi/4$. $$ \left[\tan\left(\frac{x}{2}\right)\right]_{\pi/4}^{3\pi/4} = \tan\left(\frac{3\pi/4}{2}\right) - \tan\left(\frac{\pi/4}{2}\right) $$ $$ = \tan\left(\frac{3\pi}{8}\right) - \tan\left(\frac{\pi}{8}\right) $$ Step 4: Substitute the known values and simplify. We use the known values for $\tan(3\pi/8)$ and $\tan(\pi/8)$: $\tan(3\pi/8) = 1 + \sqrt{2}$ $\tan(\pi/8) = \sqrt{2} - 1$ Substitute these values into the expression from Step 3: $$ (1 + \sqrt{2}) - (\sqrt{2} - 1) = 1 + \sqrt{2} - \sqrt{2} + 1 $$ $$ = 2 $$ The final answer is $\boxed{2}$.