I'm looking at how an integral was computed in "Paul's online math notes" in the section on U-substitution and am a bit confused. In the original expression there are $2$ $x$'s, one in front of $\cos$ and the other as the numerator in the second term. When he substitutes $xdx=\dfrac{1}{2}du$ why does $\dfrac{1}{2}du$ replace both $x$ terms?
thanks
They are factoring the $x$ out of both terms so there will only be one $x$ in front of parenthesis.
It is just a matter of convenience. You do not have to factor if you do not like it. Then of course you have two $ xdx$ to deal with.