For example, how would I know to make my $u$ the $\sqrt{1-x}$ when given a problem such as to evaluate the integral of $x^2\sqrt{1-x}$? Is there a logical thinking you use to know $u = \sqrt{1-x}$ rather than just $1-x$? I just learned $u$-substitution the other day, but my teacher stressed that in nearly all cases we will not have a u raised to a power.
Any info on this would be grateful!