Ok thanks, that's making more sense now. I'm still a little confused though as to why you can actually do this whole substitution of 'u' for 'x' in this specific case. In all the others problems I do where you need to use substitution, there is an obvious 'parent function' in the integral (which you can substitute for u), and the derivative of the parent function (which you can substitute for du/dx). In this problem though, this form of their being a 'parent function' is not evident. What part of the 'formula' for substitution is being applied here such that it is legal?
In any integral, can you simply do as you are saying an just substitute all the x terms for u, including the dx itself? Is this the over-arching idea of substitution? In what I have learnt so far, you substitute in 'u' for the parent function and then du/dx for the derivative of that parent function.. then the dx's simply cancel. Is this a bad way to think of it?
I'm not familiar with the use of the term "parent function" this way; but it is true that commonly we
recognize a substitution by seeing some function within the integrand, whose derivative is also present (such as an x^2 somewhere, together with 2x being multiplied). But the reason for substitution does not have to be so obvious. Substitution just means
substituting u (or whatever name you like) for some expression, and du for dx, regardless of what else is involved. Quite often something has to be modified in order for the derivative to be there.
This example is relatively subtle; I might not recognize quickly what to do. In order to see the
possibility that u = x^(1/6) will be helpful, you need to see that there are two radicals with different indices, and that these can be seen as the cube and the square, respectively, of the sixth root. Whenever some little function like that appears repeatedly, it is a candidate for substitution. The derivative of u is not present in this case, but I would just try the substitution in order to see what the result would be -- I don't expect to know ahead of time what will necessarily happen, but just try something out of hope, since I don't see anything else likely to work.
In a sense, the substitution here was done in
reverse compared to the simple examples you probably saw first. Rather than replace a function of x with u, and du/dx dx with du, they
replace x itself with u^6 and dx with dx/du du. The effect is the same. I notice that
Wikipedia gives two examples, which they call right-to-left and left-to-right. The second example may be beyond your current knowledge in detail, but you can see that, like here, they replace x and dx, rather than u and du. This style of substitution is probably rarer, at least in your textbook, but is quite powerful.
Having seen this example, you now have that much more experience by which you can think about possibilities in the future. That is how methods of integration tend to work. I'm guessing you were given it to introduce you to this style of substitution.