<s>[INST] Finding the indefinite integral of a root function I'm stuck on a particular integral problem.  The problem is stated as:
$$\int x \sqrt{2x+1} dx$$
My working thus far:
$$\int x \sqrt{2x+1} dx = \frac{1}{2}x^2\frac{2}{3}(2x+1)^\frac{3}{2}$$
$$\frac{1}{3}x^2(2x+1)\frac{3}{2}$$
...
I think I am totally wrong here and completely on another course.  Will somebody please point out where I'm screwing up, and possibly guide me to the correct answer?
Thanks!
 [/INST] It looks like to integrate the product, you found the product of the integration of each factor. We can't do that! 
If $\int f(x) \,dx = F(x)$ and $\int g(x)\,dx = G(x)$

$$\int f(x)\cdot g(x) \,dx \neq F(x)G(x) + C$$


Let's start over.
Note that since the integrand is defined only for $2x+1\geq0$, we put
$$\underbrace{u^2 = 2x+1}_{u = \sqrt{2x+1}} \implies 2u\,du = 2\,dx\iff u\,du = dx$$  
Note also that we also have $2x = u^2 -1 \iff x = \frac 12( u^2 - 1)$.
Then $$I = \int x \sqrt{2x+1} dx = \frac 12\int (u^2 -1)u\cdot u \,du =\frac 12\int(u^4 - u^2)\,du$$
$$\begin{align} I & = \frac 12\left( \frac {u^5}{5} - \frac{u^3}{3}\right) + C \\
& = \frac 12 \left(\frac{(2x + 1)^{5/2}}{5} - \frac{(2x+1)^{3/2}}{3}\right) + C\end{align}$$
</s>