I am a little confused here, how does removing $1/2$ from the function to the outside of the integral get rid of the $t$ in the numerator in this problem?
$\eqalign{ \int\dfrac t{t^4+25}dt & = \dfrac12\int\dfrac{1}{(t^2)^2+5^2}(2)\,dt \\ &= \dfrac1{10}\arctan\left(\dfrac{t^2}5\right)+C }$