Question

Calcule a seguinte integral

Answer 1

$\displaystyle \int\limits^3_1\frac{\text{dx}}{\sqrt{3-\text x}} \\\\ \underline{\text{fa{\c c}amos}}:\\\\ \text u = 3-\text x \to -\text {du}= \text{dx} \\\\\underline{\text{substituindo}}: \\\\ \int\limits^3_1 \frac{-\text{du}}{\sqrt{\text u}}\to -\int\limits^3_1 \ \text u^{\displaystyle (\frac{-1}{2})}\text{du}$

$\displaystyle -\left[\begin{array}{c} \frac{\displaystyle \text u ^{\displaystyle (\frac{-1}{2}+1)}}{\displaystyle \frac{-1}{2}+1} \end{array}\right]\limits^3_1$

$\displaystyle -\left[\begin{array}{ccc}2.\text u^{\displaystyle (\frac{1}{2})} \end{array}\right] \limits^3_1 \to -\left[\begin{array}{ccc}2.(3-\text x)^{\displaystyle (\frac{1}{2})} \end{array}\right] \limits^3_1 \\\\\\\ -\left[\begin{array}{ccc}2.\sqrt{(3-\text x)} \end{array}\right] \limits^3_1 \\\\\\ \underline{\text{Aplicando os limites de integra{\c c}{\~a}o}}: \\\\\\ -[2\sqrt{3-3}\ -\ 2\sqrt{3-1}] \to 0 +2\sqrt{2}$

Portanto :

$\displaystyle \huge\boxed{\int\limits^3_1 \frac{\text{dx}}{\sqrt{3-\text x}}=2\sqrt{2}\ }\checkmark$

Letra D