Why use u-substitution on this problem?
I suppose if you HAVE to use u-substitution, you could let instead.
That would give you the following integral:
One note: When using u-substitution, always look for the derivative of u (the one with the original variable) to cancel out. In this case, you identified the problem with the substitution you originally did - couldn't cancel out the t's. But with the derivative would include a 't' which WOULD cancel out.