<s>[INST] Subsitution in Integrals Okay, so I'm working on definite integrals and I calculated the indefinte integral of $\frac{x}{\sqrt{x+1}}$ to be $\frac{2}{3}u^{3/2}-2\sqrt{u}$ where $u=x+1$. The definite integral is on the interval $[0,5]$ so I used my $u$-subsitution equation on the interval $[1,6]$. I keep getting $7.9981$ but I know that's incorrect. What am I doing wrong?
 [/INST] Same answer, but with the useful trick of "adding $0$":
\begin{align}
\int \frac{x}{\sqrt{x+1}}dx = \int \frac{x+1 -1}{\sqrt{x+1}}&= \int \left(\frac{x+1}{\sqrt{x+1}} - \frac{1}{\sqrt{x+1}} \right)dx
\\
&= \int\left( (x+1)^{\frac{1}{2}} -  (x+1)^{- \frac{1}{2}}\right)dx
\\
&= \frac{2}{3} (x+1)^{\frac{3}{2}} - 2 (x+1)^{\frac{1}{2}}.
\end{align}
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