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I am attempting to learn about the substitution rule and I can't make sense of what Stewart is trying to say. "To find this integral we use the problem solving strategy of introducing something extra. Here the something extra is a new variable, we change from the variable x to a new variable u. Suppose that we let u be the quantity under the root sing 1, $u=1+x^2$ Then the differential of u is du=2xdx. Notice that if the dx in the notation for an integral were to be interpreted as a differnetial then the differential 2xdx would occur in 1 and so formally without justifying our calculation we could write" The rest doesn't really matter, I just don't understand what is going on at all.

The differential of u is $.5(1+x^2)(2x)$ not what he has.

1: $\int 2x \sqrt{1+x^2}$

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    How do you get that the differential of $u$ is $.5(1+x^2)2x$? If $u=1+x^2$, then $du = d(1+x^2) = d(1)+d(x^2) = 0 + 2x\,dx$, exactly what Stewart says.2011-11-08
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    What does du=2xdx mean? The differnetial of du with respect to u is equal to 2x differential of dx with respect to x?2011-11-08
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    It means that if $u=1+x^2$, then "a small change in $u$" (the differential) will necessarily be the same size as a "small change in $x$" (the differential in $x$) times twice the value of $x$ you are in. But I repeat my question: why is it that you think that "the differential of $u$ is $0.5(1+x^2)(2x)$." In your mind, what does *that* mean and *why* is that what the differential of $u$ "is"?2011-11-08
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    I was reading it too quickly and took the differential of the square root of u not u. I really do not know what you are talking about small changes, is this something to do with reimann sums?2011-11-08
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    Do you remember what a differential is? If you don't know what a differential is, then of course you can't make sense of any explanation for why they are what they are. I don't know if [this](http://math.stackexchange.com/questions/23902/what-is-the-practical-difference-between-a-differential-and-a-derivative/23914#23914) will help.2011-11-08
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    Off the top of my head a differential is the small difference in a function.2011-11-08
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    I have read that already and I can make sense of it a little bit but I always forget.2011-11-08
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    @Jordan Might I suggest you take a look at The Calculus Lifesaver videos - http://press.princeton.edu/video/banner/ - specifically video 12 addresses u-substitution.2011-11-09

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