Question

₀Calcular as integrais duplas:

∫₁²∫₀¹yeˣУdxdy

Answer 1

$\mathrm{\int\limits_1^2\int\limits_0^1ye^{xy}dxdy}\\\\\\ \mathbf{Resolvendo\ \int\limits_0^1ye^{xy}\ dx:}\\\\ \mathrm{u=xy\ \ \|\ \ \dfrac{du}{dx}=y\ \to\ \dfrac{du}{y}=dx}\\\\ \mathrm{\int\limits_0^1y\dfrac{e^u}{y}\ du=\bigg(\int e^u\ du\bigg)\bigg|_{x=0}^1=e^{xy}\bigg|_{x=0}^1=e^{1.y}-e^{0.y}=\boxed{\mathrm{e^y-1}}}\\\\\\ \mathbf{Resolvendo\ \int\limits_1^2e^y-1\ dy:}\\\\ \mathrm{\bigg(\int e^y\ dy-\int1\ dy\bigg)\bigg|_{y=1}^2=\bigg(e^y-y\bigg)\bigg|_{y=1}^2=e^2-2-(e-1)=}\\\\ \mathrm{=e^2-2-e+1=e^2-e-1=\boxed{\mathbf{(e-1)e-1}}}$