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Chapter 13: Problem 54

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

Expert verified
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

01

Express the Integral

The initial expression of the problem is provided in the form \[ \int_{0}^{4} \int_{0}^{y} \frac{2}{(x+1)(y+1)} d x d y \]
02

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.
03

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.
04

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.

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