Find the definite integral using substitution

Suppose I have this function where a = 0 and b = .5 not sure on the latex tags for that

$\displaystyle I(x) = \int \sqrt{t^4+1}$

and I let

$\displaystyle u = t^4+1$

and

$\displaystyle du = 4t^3dx$

this is the part where I get confused...

Do I

$\displaystyle \frac{1}{4}du = t^3dx$

or solve for dx completely

If I do it at the step above and continue on to

$\displaystyle \int u^\frac{1}{2} * du$

and plug in the corresponding values my answers do not come out correctly