Are we allowed to evaluate a Doubly Improper Integral without splitting it up?

∫dx/√ x(x+1) with a lower bound of 0 and an upper bound at positive infinity.

In this definite integral f(x) has an infinite discontinuity at x = 0, so we must evaluate it as c Lim c ->0+ on the lower bound. Our upper bound is infinity so we must evaluate it as b Lim b -> inf, so both our bounds are improper. In the passage the author chose to split the integral so we have an integral bound from 0 to 1 plus another bound from 1 to positive infinity. My question is can we keep it all under one integral and evaluate both limits at the same time?