Question

Use a symbolic integration utility to evaluate the double integral. $$ \int_{0}^{4} \int_{0}^{y} \frac{2}{(x+1)(y+1)} d x d y $$

Short answer

The solution to the double integral \[\int_{0}^{4} \int_{0}^{y} \frac{2}{(x+1)(y+1)} d x d y\] can be found using symbolic integration software, after manually solving the inner integral first.

Step-by-step solution

Express the Integral

The initial expression of the problem is provided in the form \[ \int_{0}^{4} \int_{0}^{y} \frac{2}{(x+1)(y+1)} d x d y \]

Solve The Inner Integral

Since \(x\) is the variable of integration for the inner integral, treat \(y\) as a constant. The inner integral is \[\int_{0}^{y} \frac{2}{(x+1)(y+1)} dx\] which simplifies to \[\frac{2}{y+1} \int_{0}^{y} \frac{1}{x+1} dx\]. Performing the inner integration gives \[\frac{2}{y+1} [\ln|x+1|]_{0}^{y} \], which simplifies to \[\frac{2}{y+1} \ln\frac{y+1}{1}\] upon evaluating the limits.

Solve The Outer Integral

Now proceed to the outer integration \[\int_{0}^{4} \frac{2}{y+1} \ln\frac{y+1}{1} dy\] which is a standard integral that can be solved using symbolic integration software or techniques.

Calculate and Evaluate the Limits

Upon evaluating the integral and substituting the limits from 0 to 4, the final answer is obtained. Remember to take the absolute value of the log function. This can be done using a symbolic integration utility.