  We study the extension of a Lie algebroid by a representation up to homotopy,
including semidirect products of a Lie algebroid with such representations. The
extension results in a higher Lie algebroid. We give exact Courant algebroids
and string Lie 2-algebras as examples of such extensions. We then apply this to
obtain a Lie 2-groupoid integration of an exact Courant algebroid.
