Question

Resolva a integral dupla. $\int\limits^2_1 { \int\limits^1_0 {(x+y) ^- ^2 dx } \, dy$

Answer 1

$\displaystyle\int\limits_{1}^{2}\int\limits_{0}^{1}{(x+y)^{-2}\,dx\,dy}\\ \\ \\ =\int\limits_{1}^{2}{\left.-(x+y)^{-1}\right|_{0}^{1}\,dy}\\ \\ \\ =\int\limits_{1}^{2}{[-(1+y)^{-1}+(0+y)^{-1}]\,dy}\\ \\ \\ =\int\limits_{1}^{2}{\left(-\dfrac{1}{1+y}+\dfrac{1}{y} \right )dy}\\ \\ \\ =\left[-\mathrm{\ell n}(1+y)+\mathrm{\ell n}(y) \right ]_{1}^{2}\\ \\ =\left[-\mathrm{\ell n}(1+2)+\mathrm{\ell n}(2) \right ]-\left[-\mathrm{\ell n}(1+1)+\mathrm{\ell n}(1) \right ]\\ \\ =-\mathrm{\ell n}(3)+\mathrm{\ell n}(2)+\mathrm{\ell n}(2)-\mathrm{\ell n}(1)\\ \\ =\mathrm{\ell n}\left(\dfrac{2\cdot 2}{3}\right)-0\\ \\\ \\ =\mathrm{\ell n}\left(\dfrac{4}{3}\right)$