Question

Calcule a seguinte integral imprópria

Answer 1

$\displaystyle \int\limit^{\infty}_4 \ \text e^{-3\text x}\text{dx}$

já que o infinito não é um número definido, vamos aplicar o seguinte limites:

$\displaystyle \lim_{\displaystyle \text b \to \infty} \ \int\limits^{\text b}_4 \ \text e^{-3\text x}\text {dx} \\\\\\ \lim_{\displaystyle \text b \to \infty } \left[\begin{array}{c}\displaystyle \frac{-\text{e}^{-3\text x}}{3}\end{array}\right]\limits^{\text b} _4 \\\\\\ \lim_{\displaystyle \text b \to \infty} \left[\begin{array}{c}\displaystyle \frac{-1}{3.\text{e}^{3.\text b} }\end{array}\right] - \frac{-1}{3.\text{e}^{3.\text 4}}$

$\displaystyle \frac{-1}{\text e^{(\infty)}} -\frac{-1}{3.\text e^{12}} = 0 + \frac{\text e^{-12}}{3} \\\\\ \underline{\text{Portanto}}: \\\\\\ \huge\boxed{\int\limits^{\infty}_4 \ \text e^{-3\text x}\text{dx} = \frac{\text e^{-12}}{3}\ }\checkmark$

letra B