integration by substitution I am trying to unravel the question integration below. But cannot see where the 1/3 comes from where the values for u and du are substitute back in. I assume it has somthing to do with the x^2 but cant see it. 
Any advice on how to proceed or good articles on the subject would be greatly appreciated 

 A: It is because of the substitution $du = 3x^2dx$. If you notice in your integral you simply have the quantity $x^2dx$, not $3x^2dx$. So by the first equation, you can divide by three to get $x^2dx = \frac{du}{3}$. Now transforming your integral to one in terms of $u$ yields $$\begin{align}\int x^2\sqrt{x^3+2}dx = \int \sqrt{x^3+2}(x^2dx) \\ = \int \sqrt{u}\left(\frac{du}{3}\right) \\ = \frac{1}{3} \int \sqrt{u}\space du \end{align}$$ Which has a very clear antiderivative. The coefficients and constants that appear in integration can be a bit tricky to rationalize sometimes, but here's one way to think about it: What coefficient $C$ would I need on $(x^3+2)^{3/2}$ so that if I took the derivative, the coefficient would disappear? The answer here happens to be $\frac{2}{9}$, as you can see by evaluating $$\begin{align}\left[\frac{2}{9}(x^3+2)^{3/2}+\text{constant}\right]' = \frac{2}{9} \cdot 3x^2 \cdot \frac{3}{2} (x^3+2)^{1/2} \\ = \frac{6}{9} \cdot \frac{3}{2}x^2(x^3+2)^{1/2} \\ = x^2(x^3+2)^{1/2}\end{align}$$ returning you to the quantity you were trying to integrate in the first place. So, in the process of evaluating the integral, you have to in a sense "uncover" the coefficients that had cancelled out in the process of taking the derivative.
A: $$
u = x^3 +2\implies du = 3x^2dx \tag{*}
$$
Since to maintain the original integral we only need $x^2dx$ we require to divide the both sides of the final relation in eq (*) by 3.
