Question

Marque a alternativa que corresponde ao valor da integral a seguir:

Alguém pode me ajudar expondo os cálculos?

Answer 1

Olá


$\displaystyle \mathsf{\int^e_1 \left(2\ell n x~-~ \frac{4}{x} \right)dx}\\\\\\\\\boxed{\mathsf{\int \delta\ell n x=\delta(x\ell nx -x)}}\quad\delta \in \Re\qquad\qquad\Longleftarrow\text{Integral de tabela}\\\\\\\boxed{\mathsf{\int \frac{\delta}{x}dx~=~\delta\ell nx }}\quad\delta \in \Re\qquad\qquad\Longleftarrow \text{Propriedade}\\\\\\\text{Com isso tiramos que}\\\\\\\mathsf{\int^e_1 \left(2\ell n x~-~ \frac{4}{x} \right)dx}\\\\\\\mathsf{=\left(\dfrac{}{}2(x\ell nx -x)~-~4\ell n x\right)\bigg|^e_1}$


$\displaystyle \mathsf{=\left(\dfrac{}{}2((e)\ell n(e) -e)~-~4\ell n (e)\right)~-~\left(\dfrac{}{}2((1)\ell n(1) -1)~-~4\ell n (1)\right)}\\\\\\\mathsf{\ell n(1)=0}\\\mathsf{\ell n(e)=1}\\\\\\\mathsf{=\left(\underbrace{\dfrac{}{}2((e)\ell n(e) -e)~-~4\ell n (e)\right)}_{=-4}\right)~-~\left(\underbrace{\dfrac{}{}2((1)\ell n(1) -1)~-~4\ell n (1)}_{=-2}\right)}\\\\\\\\\mathsf{=-4-(-2)}\\\\\\\boxed{\mathsf{=-2}}\qquad\qquad\Longleftrightarrow\qquad\qquad\text{LETRA a)}$




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