Question:

Step 1: Define a suitable substitution and find its differential. To simplify the integrand, we make a substitution. Let $t = \tan x$. Differentiating both sides with respect to $x$, we get: $$dt = \sec^2 x \, dx$$ Step 2: Change the limits of integration. Since we are changing the variable from $x$ to $t$, we must also change the limits of integration accordingly. When $x = 0$, $t = \tan(0) = 0$. When $x = \frac{\pi}{4}$, $t = \tan\left(\frac{\pi}{4}\right) = 1$. So, the new limits of integration are from $0$ to $1$. Step 3: Substitute and evaluate the definite integral. Now, substitute $t = \tan x$, $dt = \sec^2 x \, dx$, and the new limits into the integral: $$I = \int_0^{\pi/4} \frac{\sec^2 x}{(1+\tan x)^2} dx = \int_0^1 \frac{dt}{(1+t)^2}$$ To evaluate this integral, we can write $\frac{1}{(1+t)^2}$ as $(1+t)^{-2}$. $$I = \int_0^1 (1+t)^{-2} dt$$ Integrating $(1+t)^{-2}$ with respect to $t$ gives $\frac{(1+t)^{-1}}{-1} = -\frac{1}{1+t}$. $$I = \left[-\frac{1}{1+t}\right]_0^1$$ Now, apply the limits of integration: $$I = \left(-\frac{1}{1+1}\right) - \left(-\frac{1}{1+0}\right)$$ $$I = \left(-\frac{1}{2}\right) - (-1)$$ $$I = -\frac{1}{2} + 1$$ $$I = \frac{1}{2}$$ Step 4: State the final answer. The value of the definite integral is $\frac{1}{2}$. The final answer is $\boxed{\frac{1}{2}}$.