Question

Determine o valor da integral:

a. 1/2

b. 1/6

c. 5/6

d. 8/3

e. 1

Answer 1

Resposta:  alternativa d. 8/3.

Explicação passo a passo:

Calcular a integral definida:

    $\displaystyle\int_{-1}^4\frac{x}{\sqrt{x+5}}\,dx$

Façamos a seguinte substituição:

    $x+5=u\quad\Longrightarrow\quad \left\{\begin{array}{l}x=u-5\\\\ dx=du\end{array}\right.$

Quando  $x=-1\quad\Longrightarrow\quad u=4.$

Quando  $x=4\quad\Longrightarrow\quad u=9.$

Substituindo, a integral fica

    $\displaystyle=\int_4^9\frac{u-5}{\sqrt{u}}\,du\\\\\\=\int_4^9\left(\frac{u}{\sqrt{u}}-\frac{5}{\sqrt{u}}\right)du\\\\\\=\int_4^9\left(\sqrt{u}-\frac{5}{\sqrt{u}}\right)du$

    $\displaystyle=\int_4^9 \left(u^{1/2}-\frac{5}{u^{1/2}}\right)du\\\\\\=\int_4^9 (u^{1/2}-5u^{-1/2})\,du$

    $=\left.\left(\dfrac{u^{(1/2)+1}}{\frac{1}{2}+1}-5\cdot \dfrac{u^{(-1/2)+1}}{-\frac{1}{2}+1}\right)\right|_4^9\\\\\\=\left.\left(\dfrac{u^{3/2}}{\frac{3}{2}}-5\cdot \dfrac{u^{1/2}}{\frac{1}{2}}\right)\right|_4^9\\\\\\=\left.\left(\dfrac{2}{3}\,u^{3/2}-5\cdot 2u^{1/2}\right)\right|_4^9\\\\\\=\left.\left(\dfrac{2}{3}\,u^{3/2}-10u^{1/2}\right)\right|_4^9$

    $=\left(\dfrac{2}{3}\cdot 9^{3/2}-10\cdot 9^{1/2}\right)-\left(\dfrac{2}{3}\cdot 4^{3/2}-10\cdot 4^{1/2}\right)\\\\\\=\left(\dfrac{2}{3}\cdot \sqrt{9^3}-10\cdot \sqrt{9}\right)-\left(\dfrac{2}{3}\cdot\sqrt{4^3}-10\cdot \sqrt{4}\right)\\\\\\=\left(\dfrac{2}{3}\cdot 27-10\cdot 3\right)-\left(\dfrac{2}{3}\cdot 8-10\cdot 2\right)\\\\\\=(18-30)-\left(\dfrac{16}{3}-20\right)$

    $=-\,12-\dfrac{16}{3}+20\\\\\\=\dfrac{-\,36-16+60}{3}\\\\\\=\dfrac{-\,52+60}{3}\\\\\\=\dfrac{8}{3}\quad\longleftarrow\quad \mathsf{resposta:~alternativa~d.}$

Dúvidas? Comente.

Bons estudos!