- #1

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## Homework Statement

## Homework Equations

## The Attempt at a Solution

I don't understand what exactly is going on here. They let [itex]u=(1+x^{2})[/itex], so that leaves them with this:

[tex]\int \frac{x}{u^{2}}dx[/tex]

The derivative of [itex](1+x^{2})[/itex] is simply [itex]2x[/itex]. And so:

[tex]\frac{du}{dx} = 2x \rightarrow du = 2xdx \rightarrow dx=\frac{du}{2x}[/tex]

So now, substituting in my new dx, I get:

[tex]\int \frac{x}{u^{2}2x}du[/tex]

So, is that x simply canceling out here? Is that the idea?

Which leaves me with:

[tex]\int \frac{u^{-2}}{2}du[/tex]

[tex]-\frac{1}{2u}[/tex]

Re-substituting u I get:

[tex]-\frac{1}{2(x^{2}+1)} + C[/tex]

With that being said, how do you know that the x will cancel? How are you even supposed to know that this approach will work? Is there some sort of proof to this idea, my book does not have it.