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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!

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    I somewhat prefer $1-x=u^2$.2012-02-09

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$u=\sqrt{1-x}$ is an example of a "rationalizing substitution". "Rationalizing" means getting rid of the radical. One writes $u^2=1-x$, so $2u\;du = -dx$. Then $x$ is replaced by $1-u^2$. The point is that putting the problem in a form where there are no other functions besides polynomials enables one to rely on the fact that one knows how to deal with polynomials.