$u$-substitution seems arbitrary 
Evaluate: $$ \int \frac{x^2}{\sqrt{1-x^2}}\,dx$$

The solution I came across does a $u$-substitution by letting $x = \sin(t)$. But why $\sin(t)$? It seems a lot like guessing to me - should I just guess the right $x$ for substitution? Why not $\cos(t)$? Is there any way that I could identify that "this" expression is solved by "that" trig substitution?
 A: Draw a triangle!  When looking at the term $\sqrt{1 - x^{2}}$ in the integrand, you should immediately be reminded of the Pythagorean Theorem.  If you draw a right triangle with hypotenuse length $1$, and side lengths $x$ and $\sqrt{1 - x^{2}}$, you end up with the following triangle.  I've labelled the remaining two angles $s$ and $t$.  So, if you want to isolate $x$ for a potential change of variables, based on the triangle, you can either use $\sin(s) = x$ or $\cos(t) = x$.  Either would give you a valid change of variables.

A: I think that the OP's question refers to how to distinguish why to make this change of variable. Now, I will write the answer to that question.
Suppose you want to solve an integral of the form $$\color{blue}{\boxed{\int R(x,\sqrt{a^{2}-x^{2}})dx}}$$
As in your problem, that you have $$\color{red}{\text{Example:} \quad  \int \frac{x^{2}}{\sqrt{1^{2}-x^{2}}}dx}$$So, this integral becomes a trigonometric integral with the change of variable $$\color{green}{\boxed{x=a\cos(t)}} \quad \color{green}{\boxed{x=a\sin(t)}}$$
As the MathS users indicate in the comments. Also note that you can also perform the variable change $$\color{green}{\boxed{x=a\tanh(t)}}$$ and this variable change transforms the integral into a hyperbolic integer.
