U.S. stocks mostly fell in a volatile session on Tuesday as fears that a $1 trillion bailout for Europe won't solve the region's deep-seated problems outweighed upbeat sentiment about an improving U.S. economy.

The Dow Jones industrial average <.DJI> dropped 36.88 points, or 0.34 percent, to end unofficially at 10,748.26. The Standard & Poor's 500 Index <.SPX> slipped 3.94 points, or 0.34 percent, to finish unofficially at 1,155.79. But the Nasdaq Composite Index <.IXIC> edged up 0.64 of a point, or 0.03 percent, to close unofficially at 2,375.31.

(Reporting by Rodrigo Campos; Editing by Jan Paschal)