Question

Calcular as integrais duplas:

∫₁²∫₀¹ycos(xy)dxdy

Answer 1

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