
Manipulating Fractions
I'm trying to do an intergration using the tsubstitution so I've been reading examples and I keep loosing track on a similar fraction problem. I don't understand how
$\displaystyle \frac{4}{3+(10t/(1+t^2))} * \frac{2}{1+t^2} = \frac{8}{3t^2+10t+3}$
Any insight on how you can manipulate this is greatly appreciated :)
Cheers,
Isk

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$\displaystyle {\color{red}\frac{4}{3 + \displaystyle \frac{10t}{(1+t^2)}}} \ \cdot \ {\color{blue}\frac{2}{1+t^2}}$
To multiply fractions, simply multiply numerator by numerator and denominator by denominator:
$\displaystyle = \frac{{\color{red}4} \cdot {\color{blue}2}}{{\color{red}\left(3 + \displaystyle \frac{10t}{(1+t^2)}\right)}{\color{blue}\left(1+t^ 2\right)}}$
You should know that: $\displaystyle a(b+c) = ab + ac$. Here, imagine $\displaystyle a = {\color{blue}1+t^2}$.
So, simplifying:
$\displaystyle = \frac{8}{{\color{red}3}{\color{blue}(1+t^2)} + \displaystyle \frac{{\color{red}10t}}{{\color{blue}1+t^2}} {\color{blue}\left(1+t^2\right)}}$
Notice that in the denominator, the $\displaystyle 1 + t^2$ cancels out in the second term, so we get:
$\displaystyle = \frac{8}{3(1+t^2) + 10t}$
and so on and so on ..
