Evaluate $\int ( 3x + \frac{6x^2 \sin^2(\frac{x}{2})}{x - \sin x} ) \frac{(x-\sin x)^{3/2}}{\sqrt{x}} \mathrm{d}x$ I have a really vague integration problem. It's some substitution and then integration by parts maybe. 
I got this from a friend, but it seems it's unlikely​ to be solved.


 Now my question is, how to approach such big problems?

I reduced it to $$\displaystyle{\int \bigg(3x(2x-x\cos x - \sin x)\sqrt{1-\frac{\sin x}{x}}} \bigg)\mathrm{d}x$$

How to proceed from here?
I think I got it! Consider your simplified version and try to regroup terms to make it look more organized:

$$\int 3x(2x-x\cos x-\sin x )\sqrt{1-\frac{\sin x}{x}}dx$$
$$=\int 3\sqrt{x}(x(1-\cos x) +(x-\sin x))\sqrt{x-\sin x}dx$$
Now start recognition of similar terms. Let's define
$$u=x-\sin x$$
Observe $du=(1-\cos x)dx$. The integral becomes:
$$=\int 3(x(du/dx) +u)\sqrt{ux}dx=\int 3(xdu+udx)\sqrt{ux}$$
But now you see how the parenthesis looks like a total derivative of $ux$. This leads us to the conclusion, that $u$ is not a good variable substitution, but $ux$ is! Define $t=ux=x^2-x\sin x$ and you get
$$\int 3 \sqrt{t}dt$$
You can take it from here.
Easier way to solve this problem