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Chapter 13: Problem 50

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

Short Answer

Expert verified
The detailed solution steps involve careful execution of the two levels of integration, and also simplifying expressions several times. Therefore the final result could be a simplified numeric value or an exact symbolic value, depending on the integrals.

Step by step solution

01

Perform inner integral

Perform the inner integral first using variable \(x\) and the bounds \(y\) and \(2y\): \[ \int_{y}^{2y} \ln (x+y) dx \] The antiderivative of \(\ln (x+y)\) with respect to \(x\) is \((x+y)(\ln|x+y| - 1)\). Plug in the upper and lower limit to get \( (2y+y)(\ln|2y+y| - 1) - (y+y)(\ln|y+y| - 1) \). Simplify.
02

Simplify and perform outer integral

Simplify the expression and perform the outer integral using variable \(y\) and bounds 1 and 2: \[ \int_{1}^{2} ((2y+y)(\ln|2y+y| - 1) - (y+y)(\ln|y+y| - 1)) dy \]. Simplify again.
03

Final simplification

After performing the integrations, you'll need to simplify the expression further to get a final result.

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Most popular questions from this chapter

Use a symbolic integration utility to evaluate the double integral. $$ \int_{0}^{2} \int_{x^{2}}^{2 x}\left(x^{3}+3 y^{2}\right) d y d x $$

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