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Mean Value
Attachment 25757
How does the first green box become the second green box, or more importantly WHY does the first green box become the second. I don't see any reason to remove a factor of 1/2(especially when there is no foreseeable reason to do so, I feel as if this would result in extraneous), is there some step i'm missing here? I tried partial fractions but ended up with the original equation.

Re: Mean Value
I do the problem the same way but end up with a coefficient of 1/4 instead of 1/8, the answer is close because of small decimals but still does not explain why taking the 1/2 factor to begin with was not extraneous.

Re: Mean Value
note ...
$\displaystyle \int_1^3 \frac{x}{(x^2+1)^3} \, dx = {\color{red}\frac{1}{2}} \int_1^3 \frac{{\color{red}{2}}x}{(x^2+1)^3} \, dx$
the constant 2 was inserted into the integrand to facilitate the desired substitution ... the 1/2 was placed outside the integral to compensate.