\( \int \frac { 1 + \frac { d y } { d x } } { ( x + y ) ^ { 2 } } \)

To approach the integral \( I = \int \frac { 1 + \frac { d y } { d x } } { ( x + y ) ^ { 2 } } \), we can split the integral into two parts for easier calculation: 1. The first part is \( \int \frac{1}{(x+y)^2} \, dx \). 2. The second part involves \( \int \frac{\frac{dy}{dx}}{(x+y)^2} \, dx \). For the first part, you can use a simple substitution if necessary, or process it directly based on the form. The second part requires integration by recognizing that \( \frac{dy}{dx} \) represents the derivative of \( y \) with respect to \( x \), which can be tricky if we don't know the relationship between \( x \) and \( y \). From here, you can evaluate each integral based on the specific relationship between \( x \) and \( y \) intended in your problem!