Question

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

Escolha uma:
a. 2
b. 1
c. -1
d. -2
e. 0

Answer 1

Olá Hingrid!!

Apliquemos algumas propriedades antes de efectuar as contas! Veja:

$\\ \mathsf{\int_{1}^{e} \left (2 \cdot \ln x - \frac{4}{x} \right ) \ dx =} \\\\\\ \mathsf{\int_{1}^{e} 2 \cdot \ln x \ dx - \int_{1}^{e} \frac{4}{x} \ d =} \\\\\\ \mathsf{2 \cdot \underbrace{\mathsf{\int_{1}^{e} \ln x \ dx}}_{Integral \ I} - 4 \cdot \underbrace{\mathsf{\int_{1}^{e} \frac{1}{x} \ dx}}_{Integral \ II} =}$

 Feito isto, calculamos o resultado de cada integral separadamente. Segue,

Integral I: integral por partes.

$\\ \mathsf{\int_{1}^{e}\ln x\dx}=\mathsf{\int_{1}^{e}1\cdot\ln x\dx}\\\\\\ \mathsf{\begin{cases} \mathsf{f(x) = \ln x \Leftrightarrow f'(x) = \frac{1}{x} \ dx} \\ \mathsf{g'(x) = 1 \ dx \Leftrightarrow g(x) = x}\end{cases}} \\\\\\ \mathsf{Sabemos \ que: \ \boxed{\mathsf{\int f(x) \cdot g'(x) \ dx = f(x) \cdot g(x) - \int f'(x) \cdot g(x) \ dx}}. \ Com \ isso,} \\\\\\ \mathsf{\int_{1}^{e}\ln x\cdot1\dx=\left[\ln x\cdot x\right ]_{1}^{e}-\int_{1}^{e}\frac{1}{x}\cdot x\ dx}$

$\\ \mathsf{\int_{1}^{e} \ln x \ dx = (\ln e \cdot e - \ln 1 \cdot 1) - \int_{1}^{e} 1 \ dx} \\\\\\ \mathsf{\int_{1}^{e} \ln x \ dx = (1 \cdot e - 0 \cdot 1) - \left [ x \right ]_{1}^{e}} \\\\\\ \mathsf{\int_{1}^{e} \ln x \ dx = (e - 0) - \left [ e - 1 \right ]} \\\\\\ \mathsf{\int_{1}^{e} \ln x \ dx = e - e + 1} \\\\\\ \boxed{\mathsf{\int_{1}^{e} \ln x \ dx = 1}}$


 Integral II: imediata/tabela.

$\\ \mathsf{\int_{1}^{e} \frac{1}{x} \ dx = \left [ \ln x \right ]_{1}^{e}} \\\\\\ \mathsf{\int_{1}^{e} \frac{1}{x} \ dx = \left [ \ln e - \ln 1 \right ]} \\\\\\ \mathsf{\int_{1}^{e} \frac{1}{x} \ dx = 1 - 0} \\\\\\ \boxed{\mathsf{\int_{1}^{e} \frac{1}{x} \ dx = 1}}$

 Por fim, temos que:

$\\ \mathsf{2 \cdot \underbrace{\mathsf{\int_{1}^{e} \ln x \ dx}}_{Integral \ I} - 4 \cdot \underbrace{\mathsf{\int_{1}^{e} \frac{1}{x} \ dx}}_{Integral \ II} =} \\\\\\ \mathsf{2 \cdot 1 - 4 \cdot 1 =} \\\\ \mathsf{2 - 4 =} \\\\ \boxed{\boxed{\mathsf{- 2}}}$

 Espero ter ajudado!

Answer 2

Resposta:

-2

Explicação passo-a-passo: CORRIGIDO PELO AVA