Question

Poderiam me ajudar na integral por partes (x+1)*√x+2

Answer 1

Para a integral

$\displaystyle\int(x+1) \sqrt{x + 2} \: dx$

Consideremos

$u = x + 1\Rightarrow du = dx \\ dv = \sqrt{x + 2} \: dx\Rightarrow v = \frac{2 }{3} {(x + 2)}^{ \frac{3}{2} }$

Então, temos

$\displaystyle\int(x+1) \sqrt{x + 2} \: dx \\ = \dfrac{2}{3} {(x + 2)}^{ \frac{3}{2} } (x + 1) - \displaystyle\int \dfrac{2}{3} {(x + 2)}^{ \frac{3}{2} } \: dx \\ = \dfrac{2}{3} {(x + 2)}^{ \frac{3}{2} } (x + 1) - \dfrac{2}{3} \cdot\dfrac{2}{5} {(x + 2)}^{ \frac{5}{2} } \\ = \dfrac{2}{3} {(x + 2)}^{ \frac{3}{2} } (x + 1) - \dfrac{4}{15} {(x + 2)}^{ \frac{5}{2} } \\ = \frac{2}{3} {(x + 2)}^{ \frac{3}{2} } \cdot((x + 1) - \frac{2}{5} {(x + 2)}) \\ =\frac{2}{3} {(x + 2)}^{ \frac{3}{2} } \cdot(x + 1-\frac{2}{5}x-\frac{4}{5}) \\ =\frac{2}{3} {(x + 2)}^{ \frac{3}{2} } \cdot(\frac{3}{5}x+\frac{1}{5}) \\ =\frac{2}{15} {(x + 2)}^{ \frac{3}{2} } \cdot(3x+1)$