Question

Integral dupla

$\int\limits^3_0 \int\limits^1_0 {xe^x^y} \, dydx$

Answer 1

$\int_{0}^{3}\int_{0}^{1}{xe^{xy}}\,dy\,dx\\ \\ =\int_{0}^{3}{\left(\int_{0}^{1}{xe^{xy}\,dy} \right )}\,dx\\ \\ =\int_{0}^{3}{\left(x\int_{0}^{1}{e^{xy}\,dy} \right )}\,dx\\ \\ =\int_{0}^{3}{\left[\diagup\!\!\!\!x\cdot \dfrac{e^{xy}}{\diagup\!\!\!\!x} \right ]_{y=0}^{y=1}}\,dx\\ \\ =\int_{0}^{3}{\left[e^{xy} \right ]_{y=0}^{y=1}}\,dx\\ \\ =\int_{0}^{3}{\left[e^{x\,\cdot\, 1}-e^{x\,\cdot\,0} \right ]}\,dx\\ \\ =\int_{0}^{3}{\left[e^{x}-1 \right ]}\,dx\\ \\ =\left[e^{x}-x \right ]_{x=0}^{x=3}\\ \\ =\left[e^{3}-3 \right ]-\left[e^{0}-0 \right ]\\ \\ =e^{3}-3-1-0\\ \\ =e^{3}-4$