Question

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

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

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.

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.

Final simplification

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