Calcule as integrais usando o método adequado e simplifique:
\int^{3} \sqrt{8(x-2)^{6}\left(x+\frac{1}{2}\right)^{3}} d x

Passo 1

O enunciado pede para calcular a integral indefinida
\int \sqrt{8\,(x-2)^{6}\,\bigl(x+\tfrac12\bigr)^{3}}\;dx.
O conteúdo é simplificação de radicais e integração de potências. Vamos simplificar o integrando e depois usar uma substituição para integrar termo a termo.

Passo 2

Dentro da raiz temos
8\,(x-2)^{6}\,\bigl(x+\tfrac12\bigr)^{3}.
Tirando a raiz quadrada:
\sqrt{8} = 2\sqrt{2}, \quad \sqrt{(x-2)^{6}} = (x-2)^{3}, \quad \sqrt{\bigl(x+\tfrac12\bigr)^{3}} = \bigl(x+\tfrac12\bigr)^{3/2}.
Logo
\sqrt{8\,(x-2)^{6}\,(x+\tfrac12)^{3}} =2\sqrt{2}\,(x-2)^{3}\,(x+\tfrac12)^{3/2}.
A integral fica
\int 2\sqrt{2}\,(x-2)^{3}\,(x+\tfrac12)^{3/2}\;dx.

Passo 3

Fazemos a substituição
u = x + \tfrac12, \quad du = dx, \quad x-2 = u - \tfrac52.
Então
(x-2)^{3} = \bigl(u-\tfrac52\bigr)^{3}, \quad (x+\tfrac12)^{3/2} = u^{3/2},
e a integral passa a ser
\int 2\sqrt{2}\,\bigl(u-\tfrac52\bigr)^{3}\,u^{3/2}\;du.

Passo 4

Expandimos \bigl(u-\tfrac52\bigr)^{3} e multiplicamos por u^{3/2}:
\bigl(u-\tfrac52\bigr)^{3} = u^{3} - \tfrac{15}{2}u^{2} + \tfrac{75}{4}u - \tfrac{125}{8},
\bigl(u-\tfrac52\bigr)^{3}u^{3/2} = u^{9/2} - \tfrac{15}{2}u^{7/2} + \tfrac{75}{4}u^{5/2} - \tfrac{125}{8}u^{3/2}.
Logo
\int 2\sqrt{2}\Bigl(u^{9/2} - \tfrac{15}{2}u^{7/2} + \tfrac{75}{4}u^{5/2} - \tfrac{125}{8}u^{3/2}\Bigr)du.
Integramos potência a potência:
\int u^{n}du = \frac{u^{n+1}}{n+1}.
Então cada termo dá:
\int u^{9/2}du = \frac{2}{11}\,u^{11/2},\quad \int u^{7/2}du = \frac{2}{9}\,u^{9/2},\quad \int u^{5/2}du = \frac{2}{7}\,u^{7/2},\quad \int u^{3/2}du = \frac{2}{5}\,u^{5/2}.
Multiplicando por 2\sqrt{2} e pelos coeficientes:
2\sqrt{2}\,\Bigl[ \frac{2}{11}u^{11/2} - \frac{15}{2}\cdot\frac{2}{9}u^{9/2} + \frac{75}{4}\cdot\frac{2}{7}u^{7/2} - \frac{125}{8}\cdot\frac{2}{5}u^{5/2} \Bigr].

Passo 5

Simplificando os coeficientes obtemos
\frac{4\sqrt{2}}{11}\,u^{11/2} -\frac{10\sqrt{2}}{3}\,u^{9/2} +\frac{75\sqrt{2}}{14}\,u^{7/2} -\frac{25\sqrt{2}}{4}\,u^{5/2} + C.
Voltando a u = x+\tfrac12, a antiderivada final é
\boxed{ \frac{4\sqrt{2}}{11}\,(x+\tfrac12)^{11/2} -\frac{10\sqrt{2}}{3}\,(x+\tfrac12)^{9/2} +\frac{75\sqrt{2}}{14}\,(x+\tfrac12)^{7/2} -\frac{25\sqrt{2}}{4}\,(x+\tfrac12)^{5/2} + C. }

Resposta

\frac{4\sqrt{2}}{11}\,(x+\tfrac12)^{11/2} -\frac{10\sqrt{2}}{3}\,(x+\tfrac12)^{9/2} +\frac{75\sqrt{2}}{14}\,(x+\tfrac12)^{7/2} -\frac{25\sqrt{2}}{4}\,(x+\tfrac12)^{5/2} + C.

Exercícios relacionados