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?
Best Answer
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.