Question

Considere a função F(x)=∫▒〖x cosx dx〗 . Pede-se a- a integral indefinida de F

Answer 1

Boa tarde Andrea

∫ x*cos(x) dx

∫ fg = fg - ∫ g*df

f = x, df = dx.  g = sen(x) , dg = cos(x)dx

x*sen(x) - ∫ sen(x)*dx = x*sen(x) + cos(x) + C 

Answer 2

$\mathrm{F(x)=\int x\cos{x}\ dx}\\\\ \mathrm{u=x\ \to\ u'=1\ \ \|\ \ v'=\cos{x}\ \to\ v=\sin{x}}\\\\ \mathrm{\int x\cos{x}\ dx=x.\sin{x}-\int\sin{x}=x\sin{x}-(-\cos{x})}\\\\ \boxed{\mathrm{F(x)=x\sin{x}+\cos{x}+c}}\\\\ \mathrm{\int F(x)\ dx=\int x\sin{x}+\cos{x}\ dx=\int x\sin{x}\ dx+\int\cos{x}\ dx=}\\\\ \mathrm{=x(-\cos{x})-\int-\cos{x}\ dx+\sin{x}=-x\cos{x}+\sin{x}+\sin{x}}\\\\ \boxed{\mathbf{\int F(x)=2\sin{x}-x\cos{x}+cx+C}}$