<s>[INST] Changing limits of integration by parts I want to calculate the integrate of:
$$\int_0^1{x\sqrt{x+2}dx}$$
I use integration by parts:
$u=x,du=dx $ and $dv=\sqrt{x+2}dx ,v=\frac{2}{3}(x+2)^{3/2}$
$$x\frac{2}{3}(x+2)^{3/2}-\int_a^b\frac{2}3(x+2)^{3/2}dx$$
I have to change the $a$ and $b$ bounds? And if yes, when i evaluate the final result, wich bounds should i apply to the first part? The originals or the new?
 [/INST] The development i suppose is : 
\begin{equation}
\int_0^1 x \sqrt{x+2}dx = \int_0^1 x (\frac{2}{3}(x+2)^{\frac{3}{2}})'dx =
 \frac{2}{3}x(x+2)^{\frac{3}{2}} \Big|_{0}^{1} - \int_0^1 \frac{2}{3}(x+2)^{\frac{3}{2}}dx = \\ = \frac{2}{3}x(x+2)^{\frac{3}{2}} \Big|_{0}^{1} - \frac{4}{15} (x+2)^{\frac{5}{2}}\Big|_0^1 = \frac{2}{3}[3^{\frac{3}{2}} - 0] - \frac{4}{15}(3^{\frac{5}{2}}-2^{\frac{5}{2}})= \frac{2}{3}3^{\frac{3}{2}} - \frac{4}{15}(3^{\frac{5}{2}}-2^{\frac{5}{2}})
\end{equation}
</s>