Question

Evaluate (1 to 4)∫(1 to 2)∫(x/y + y/x) dydx.

Answer 1

$\displaystyle\begin{aligned} \int_1^4\int_1^2\left(\frac{x}{y} + \frac{y}{x}\right) \mathrm{d}y\,\mathrm{d}x \\&= \int_1^4\left(x\ln(y) + \frac{y^2}{2x}\right)\biggr\vert_1^2\,\mathrm{d}x\\ &= \int_1^4\left(x\ln(2) + \frac{2}{x} - \frac{1}{2x}\right)\mathrm{d}x\\ &= \int_1^4\left(x\ln(2) + \frac{3}{2x}\right)\mathrm{d}x\\&= \left(\frac{x^2\ln(2)}{2} + \frac{3}{2}\ln(x)\right)\biggr\vert_1^4 \\&= 8\ln(2) + \frac{3\ln(4)}{2} - \frac{\ln(2)}{2} \\&= 8\ln(2) + 3\ln(2) - \frac{\ln(2)}{2} = \frac{21\ln(2)}{2} \end{aligned}$